Strong Laws of Large Numbers — Examples

Variance and covariance identities for random variables: Let $X,Y$ be square-integrable real random variables on one probability space. Then $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2,$ $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\mathrm{.}$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

The Axiom of Choice: The Axiom of Choice (AC) is the following statement.

Every family of nonempty sets has a choice function (def-choice-function).

Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$.

An equivalent formulation is that a product of nonempty sets is nonempty: if $X_i\mathrm{\ne }\mathrm{\varnothing }$ for every $i\in I$, then ${\prod }_{i\in I}{X}_i\mathrm{\ne }\mathrm{\varnothing }$. Here ${\prod }_{i\in I}{X}_i$ is the set of functions $f$ with domain $I$ such that $f(i)\in X_i$ for every $i\in I$; when a family of nonempty sets is indexed by itself, such an $f$ is precisely a choice function for it.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the following statement.

For every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets indexed by $\mathbb{N}$ there is a function $f$ with domain $\mathbb{N}$ such that $f(n)\in X_n$ for every $n\in \mathbb{N}$.

Equivalently, in the vocabulary of def-choice-function: every at most countable family of nonempty sets (def-countable) has a choice function.

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain: Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when

$\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$

The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement.

For every nonempty set $X$, every relation $R$ entire on $X$, and every $a\in X$, there is a function $x:\mathbb{N}\to X$ (def-function, def-natural-numbers) with $x_0=a\text{and}{x}_{n}R{x}_{n+1}\text{ for every }n\in \mathbb{N}\mathrm{.}$

Here a sequence in $X$ means a function from $\mathbb{N}$ to $X$, not necessarily a real-valued sequence. As everywhere in this library $\mathbb{N}$ contains $0$, and the sequence is indexed from $0$; the term $x_0$ is the prescribed starting point $a$ and every later term is related to its predecessor.

What DC adds to what came before. def-choice-function and def-axiom-of-choice select one element from each member of a family that is fixed in advance, and def-countable-choice does the same for a family indexed by $\mathbb{N}$. In both, the family is given before any selection is made. DC is the principle needed when the $n$-th set to select from is not known until the first $n$ selections have been made: here the admissible values of $x_{n+1}$ are exactly the $R$-successors of $x_n$, so the family being chosen from is built along the choosing. That is precisely the situation ${\mathrm{A}\mathrm{C}}_{\omega }$ does not cover, and it is why a construction "pick $x_{n+1}$ depending on $x_n$, for every $n$ at once" is not licensed by countable choice.

The starting point may be dropped. The formally weaker statement obtained by deleting the clause $x_0=a$ — for every nonempty $X$ and every entire $R$ there is a sequence with $x_{n}R{x}_{n+1}$ for all $n$ — is an immediate consequence of the form above, since $X$ is nonempty and any of its elements may be taken as $a$. The reverse derivation is standard and is not needed anywhere in this library, so it is not carried out; every use below prescribes $x_0$.

$R$ need not be an order and the terms need not be distinct. What DC delivers is a sequence, that is a function $\mathbb{N}\to X$, not a chain in the order-theoretic sense (def-chain). The relation may be symmetric, and the sequence may repeat a value or be constant; all that is asserted is $x_{n}R{x}_{n+1}$ at every index.

Assuming countable and dependent choice, countable products of arbitrary probability spaces: Assume countable choice and dependent choice. For probability spaces $(E_n,{\mathcal{E}}_n,{\mu }_n{)}_{n\in \mathbb{N}}$ there is a unique probability measure on the canonical countable-product sigma-algebra having the prescribed finite product marginals.

Coordinate random elements of a countable product are independent: Under the measure of F6, the coordinate maps $X_n(x)=x_n$ have laws ${\mu }_n$ and are independent.

Iid finite variance strong law: IID square-integrable real variables satisfy $S_n\mathrm{/}n\to \mathbb{E}{X}_1$ almost surely.

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

The expectation of an indicator is the probability of the event: Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space and let $A\in \mathcal{F}$. Then the indicator ${\mathbf{1}}_A$ satisfies $\mathbb{E}[{\mathbf{1}}_A]=\mathbb{P}(A)\mathrm{.}$

Kolmogorov iid l1 strong law: For IID real $(X_n{)}_{n\ge 1}$ with $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$, $S_n\mathrm{/}n\to \mu =\mathbb{E}{X}_1$ almost surely.

Finite and countable subadditivity of measures: Let $\mu$ be a measure and let $(E_k{)}_{k\in \mathbb{N}}$ be measurable. Then

$\mu \left({\bigcup }_{k\in \mathbb{N}}{E}_k\right)\le {\sum }_{k=0}^{\mathrm{\infty }}\mu (E_k)\mathrm{.}$

For every $m\in \mathbb{N}$ one also has

$\mu \left({\bigcup }_{k<m}{E}_k\right)\le {\sum }_{k<m}\mu (E_k),$

including $m=0$, where both sides are $0$.

Continuity and derivatives of positive-base real powers: For $a>0$, the function $x\mapsto a^x$ is continuous on $\mathbb{R}$ and $(a^x{)}^{\mathrm{\prime }}=a^x\log a\mathrm{.}$ For $\alpha \in \mathbb{R}$, the function $x\mapsto x^{\alpha }$ is continuous and differentiable on $(0,\mathrm{\infty })$, with $(x^{\alpha }{)}^{\mathrm{\prime }}=\alpha x^{\alpha -1}\mathrm{.}$

Probability laws correspond to distribution functions: Assume the Axiom of Countable Choice.

1. Let $X$ be a real random variable, let ${\mathbb{P}}_X$ be its law, and let $F_X(x)=\mathbb{P}(X\le x)$. Then $F_X$ is nondecreasing and right-continuous, satisfies ${\lim }_{x\to -\mathrm{\infty }}{F}_X(x)=0,{\lim }_{x\to +\mathrm{\infty }}{F}_X(x)=1,$ and obeys ${\mathbb{P}}_X((a,b])=F_X(b)-F_X(a)(a<b)\mathrm{.}$
2. Conversely, if $F:\mathbb{R}\to \mathbb{R}$ is nondecreasing and right-continuous with ${\lim }_{x\to -\mathrm{\infty }}F(x)=0,{\lim }_{x\to +\mathrm{\infty }}F(x)=1,$ then there is a unique Borel probability measure $\mu$ on $\mathbb{R}$ such that $\mu ((a,b])=F(b)-F(a)(a<b),$ equivalently $F(x)=\mu ((-\mathrm{\infty },x])(x\in \mathbb{R})\mathrm{.}$

The indefinite integral of a nonnegative measurable function is a measure: Let $f:X\to [0,+\mathrm{\infty }]$ be measurable and define ${\nu }_f(A):={\int }_{A}fd\mu (A\in \mathcal{A})\mathrm{.}$ Then ${\nu }_f$ is a measure on $(X,\mathcal{A})$.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: Let $a<b$ be reals and let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ (def-continuity-real). Then $f$ is bounded (def-bounded-set) and Riemann integrable on $[a,b]$ (def-darboux-integral).

The proof gives more than integrability: it gives a partition that works. For every real $\epsilon >0$ the uniform partition into $N$ parts already satisfies $U(f,P)-L(f,P)<\epsilon$, as soon as $N$ is large enough that $(b-a)\mathrm{/}N$ is below the $\delta$ that uniform continuity supplies for $\epsilon \mathrm{/}(2(b-a))$. Uniform continuity is exactly what makes one $\delta$ serve all $N$ subintervals at once, and it is the only place where the compactness of $[a,b]$ is used.

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: Let $a<b$ be reals, let $G:[a,b]\to \mathbb{R}$ be differentiable at every point of $[a,b]$ as a function on $[a,b]$ (def-derivative; at $a$ and $b$ this is the one-sided derivative), let $f:=G^{\mathrm{\prime }}$, and suppose $f$ is integrable on $[a,b]$ (def-darboux-integral). Then

${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$

Both hypotheses are needed and neither is removable. A function may be differentiable everywhere with $G^{\mathrm{\prime }}$ not integrable — then the left-hand side does not exist (an everywhere differentiable function with unbounded derivative) — and an integrable $f$ need not be the derivative of anything (the sign function); both witnesses are on the companion page.

No continuity of $f$ is assumed, which is what makes this the working form: the theorem evaluates ${\int }_a^{b}f$ for every integrable derivative, not only for continuous integrands.

A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: Assume the Axiom of Countable Choice. Let $a<b$ and let $f:[a,b]\to \mathbb{R}$ be bounded and Riemann integrable. Then $f$ is Lebesgue measurable on $[a,b]$ and is integrable there, and its Lebesgue integral equals its Riemann integral: ${\int }_{[a,b]}fd{\lambda }_1={\int }_a^{b}f(x)dx\mathrm{.}$

This is the point at which the completeness of Lebesgue measure is used essentially: the proof obtains a Borel function equal to $f$ almost everywhere, and measurability of $f$ itself is then a completeness statement.

Monotone convergence for the integral: Let $0\le f_1\le f_2\le \cdots$ be measurable and suppose $f_n(x)\uparrow f(x)$ for every $x$. Then $\int f_{n}d\mu \uparrow \int fd\mu \mathrm{.}$

Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system: Let $\mathcal{P}$ be a $\pi$-system on $X$ generating $\mathcal{A}$, and let $\mu ,\nu$ be measures on $(X,\mathcal{A})$ that agree on $\mathcal{P}$. Suppose there is an increasing sequence $(P_n)$ in $\mathcal{P}$ with

$X={\bigcup }_n{P}_n,\mu (P_n)=\nu (P_n)<+\mathrm{\infty }(n\in \mathbb{N})\mathrm{.}$

Then $\mu =\nu$ on $\mathcal{A}$.

Layer-cake formulas for random variables: Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space.

1. If $X:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ is measurable, then $\mathbb{E}[X]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X>t)dt,$ where the right-hand side may be $+\mathrm{\infty }$.
2. If $X$ is an integrable real random variable, then $\mathbb{E}[X]={\int }_0^{\mathrm{\infty }}\mathbb{P}(X>t)dt-{\int }_0^{\mathrm{\infty }}\mathbb{P}(X<-t)dt\mathrm{.}$

For 0 < p < infinity, the layer-cake formula computes the integral of |f|^p from the distribution function: Let $(X,\mathcal{A},\mu )$ be a measure space, let $f:X\to \mathbb{C}$ be measurable, and let $0<p<\mathrm{\infty }$. Then ${\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu =p{\int }_0^{\mathrm{\infty }}{t}^{p-1}\mu (\{\mathrm{\mid }f\mathrm{\mid }>t\})dt=p{\int }_0^{\mathrm{\infty }}{t}^{p-1}{A}_f(t)dt,$ where either side may be $+\mathrm{\infty }$.

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain: Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when

$\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$

The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement.

For every nonempty set $X$, every relation $R$ entire on $X$, and every $a\in X$, there is a function $x:\mathbb{N}\to X$ (def-function, def-natural-numbers) with $x_0=a\text{and}{x}_{n}R{x}_{n+1}\text{ for every }n\in \mathbb{N}\mathrm{.}$

Here a sequence in $X$ means a function from $\mathbb{N}$ to $X$, not necessarily a real-valued sequence. As everywhere in this library $\mathbb{N}$ contains $0$, and the sequence is indexed from $0$; the term $x_0$ is the prescribed starting point $a$ and every later term is related to its predecessor.

What DC adds to what came before. def-choice-function and def-axiom-of-choice select one element from each member of a family that is fixed in advance, and def-countable-choice does the same for a family indexed by $\mathbb{N}$. In both, the family is given before any selection is made. DC is the principle needed when the $n$-th set to select from is not known until the first $n$ selections have been made: here the admissible values of $x_{n+1}$ are exactly the $R$-successors of $x_n$, so the family being chosen from is built along the choosing. That is precisely the situation ${\mathrm{A}\mathrm{C}}_{\omega }$ does not cover, and it is why a construction "pick $x_{n+1}$ depending on $x_n$, for every $n$ at once" is not licensed by countable choice.

The starting point may be dropped. The formally weaker statement obtained by deleting the clause $x_0=a$ — for every nonempty $X$ and every entire $R$ there is a sequence with $x_{n}R{x}_{n+1}$ for all $n$ — is an immediate consequence of the form above, since $X$ is nonempty and any of its elements may be taken as $a$. The reverse derivation is standard and is not needed anywhere in this library, so it is not carried out; every use below prescribes $x_0$.

$R$ need not be an order and the terms need not be distinct. What DC delivers is a sequence, that is a function $\mathbb{N}\to X$, not a chain in the order-theoretic sense (def-chain). The relation may be symmetric, and the sequence may repeat a value or be constant; all that is asserted is $x_{n}R{x}_{n+1}$ at every index.

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

Kolmogorov iid l1 strong law: For IID real $(X_n{)}_{n\ge 1}$ with $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$, $S_n\mathrm{/}n\to \mu =\mathbb{E}{X}_1$ almost surely.

Variance and covariance identities for random variables: Let $X,Y$ be square-integrable real random variables on one probability space. Then $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2,$ $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\mathrm{.}$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents: For $a,b>0$ and $r,s\in \mathbb{R}$, $a^{r+s}=a^r{a}^s,(ab{)}^r=a^r{b}^r,(a\mathrm{/}b{)}^r=a^r\mathrm{/}{b}^r,(a^r{)}^s=a^{rs}\mathrm{.}$

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$, ${\sum }_{k\ge 1}\frac{1}{k^p}\text{ converges}⟺p>1.$

Strong law under summable normalized variances: Let $(X_n{)}_{n\ge 1}$ be independent square-integrable real random variables. Let $0<b_n$ be deterministic and nondecreasing with $b_n\to \mathrm{\infty }$. If ${\sum }_{n\ge 1}\frac{\mathrm{Var}(X_n)}{b_n^2}<\mathrm{\infty },$ then $\frac{1}{b_n}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)⟶0\text{almost surely}\mathrm{.}$ In particular, for IID centered square-integrable variables and any $\epsilon >0$, $S_n\mathrm{/}[\sqrt{n}(\log n{)}^{1\mathrm{/}2+\epsilon }]\to 0$ almost surely (the displayed normalization is used for $n\ge 2$).

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

Second Borel-Cantelli lemma under pairwise independence: Let $(A_n{)}_{n\in \mathbb{N}}$ be pairwise independent events with ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)=+\mathrm{\infty }\mathrm{.}$ Then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=1.$

Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series: For every $x\in \mathbb{R}$,

$\frac{d}{dx}\arctan x=\frac{1}{1+x^2},\arctan x={\int }_0^x\frac{dt}{1+t^2}\mathrm{.}$

For $\mathrm{\mid }x\mathrm{\mid }<1$,

$\arctan x={\sum }_{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{2n+1}\mathrm{.}$

At the endpoint, the ordinarily convergent alternating series satisfies

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \mathrm{.}$

Probability laws correspond to distribution functions: Assume the Axiom of Countable Choice.

1. Let $X$ be a real random variable, let ${\mathbb{P}}_X$ be its law, and let $F_X(x)=\mathbb{P}(X\le x)$. Then $F_X$ is nondecreasing and right-continuous, satisfies ${\lim }_{x\to -\mathrm{\infty }}{F}_X(x)=0,{\lim }_{x\to +\mathrm{\infty }}{F}_X(x)=1,$ and obeys ${\mathbb{P}}_X((a,b])=F_X(b)-F_X(a)(a<b)\mathrm{.}$
2. Conversely, if $F:\mathbb{R}\to \mathbb{R}$ is nondecreasing and right-continuous with ${\lim }_{x\to -\mathrm{\infty }}F(x)=0,{\lim }_{x\to +\mathrm{\infty }}F(x)=1,$ then there is a unique Borel probability measure $\mu$ on $\mathbb{R}$ such that $\mu ((a,b])=F(b)-F(a)(a<b),$ equivalently $F(x)=\mu ((-\mathrm{\infty },x])(x\in \mathbb{R})\mathrm{.}$

The recursion theorem: Let $(N,0,\sigma )$ be a Peano system (def-peano-system), in particular the natural numbers $\mathbb{N}$ (def-natural-numbers). For any set $A$, any element $a\in A$, and any function $f:A\to A$, there is a unique function $g:N\to A$ such that $g(0)=a$ and $g(\sigma (n))=f(g(n))$ for all $n\in N$.

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: Let $a<b$ be reals and let $f:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$ (def-continuity-real). Then $f$ is bounded (def-bounded-set) and Riemann integrable on $[a,b]$ (def-darboux-integral).

The proof gives more than integrability: it gives a partition that works. For every real $\epsilon >0$ the uniform partition into $N$ parts already satisfies $U(f,P)-L(f,P)<\epsilon$, as soon as $N$ is large enough that $(b-a)\mathrm{/}N$ is below the $\delta$ that uniform continuity supplies for $\epsilon \mathrm{/}(2(b-a))$. Uniform continuity is exactly what makes one $\delta$ serve all $N$ subintervals at once, and it is the only place where the compactness of $[a,b]$ is used.

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: Let $a<b$ be reals, let $G:[a,b]\to \mathbb{R}$ be differentiable at every point of $[a,b]$ as a function on $[a,b]$ (def-derivative; at $a$ and $b$ this is the one-sided derivative), let $f:=G^{\mathrm{\prime }}$, and suppose $f$ is integrable on $[a,b]$ (def-darboux-integral). Then

${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$

Both hypotheses are needed and neither is removable. A function may be differentiable everywhere with $G^{\mathrm{\prime }}$ not integrable — then the left-hand side does not exist (an everywhere differentiable function with unbounded derivative) — and an integrable $f$ need not be the derivative of anything (the sign function); both witnesses are on the companion page.

No continuity of $f$ is assumed, which is what makes this the working form: the theorem evaluates ${\int }_a^{b}f$ for every integrable derivative, not only for continuous integrands.

A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: Assume the Axiom of Countable Choice. Let $a<b$ and let $f:[a,b]\to \mathbb{R}$ be bounded and Riemann integrable. Then $f$ is Lebesgue measurable on $[a,b]$ and is integrable there, and its Lebesgue integral equals its Riemann integral: ${\int }_{[a,b]}fd{\lambda }_1={\int }_a^{b}f(x)dx\mathrm{.}$

This is the point at which the completeness of Lebesgue measure is used essentially: the proof obtains a Borel function equal to $f$ almost everywhere, and measurability of $f$ itself is then a completeness statement.

Monotone convergence for the integral: Let $0\le f_1\le f_2\le \cdots$ be measurable and suppose $f_n(x)\uparrow f(x)$ for every $x$. Then $\int f_{n}d\mu \uparrow \int fd\mu \mathrm{.}$

The indefinite integral of a nonnegative measurable function is a measure: Let $f:X\to [0,+\mathrm{\infty }]$ be measurable and define ${\nu }_f(A):={\int }_{A}fd\mu (A\in \mathcal{A})\mathrm{.}$ Then ${\nu }_f$ is a measure on $(X,\mathcal{A})$.

Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system: Let $\mathcal{P}$ be a $\pi$-system on $X$ generating $\mathcal{A}$, and let $\mu ,\nu$ be measures on $(X,\mathcal{A})$ that agree on $\mathcal{P}$. Suppose there is an increasing sequence $(P_n)$ in $\mathcal{P}$ with

$X={\bigcup }_n{P}_n,\mu (P_n)=\nu (P_n)<+\mathrm{\infty }(n\in \mathbb{N})\mathrm{.}$

Then $\mu =\nu$ on $\mathcal{A}$.

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: For $x>0$, $\log$ is differentiable and ${\log }^{\mathrm{\prime }}(x)=\frac{1}{x},\log x={\int }_1^x\frac{dt}{t}\mathrm{.}$

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: The function $\log :(0,\mathrm{\infty })\to \mathbb{R}$ is continuous and strictly increasing, is onto $\mathbb{R}$, and satisfies, for $x,y>0$, $\log (xy)=\log x+\log y,\log (x\mathrm{/}y)=\log x-\log y,\log (1\mathrm{/}x)=-\log x\mathrm{.}$ Also $\log 1=0$.

Integrability is necessary for an iid finite mean strong law: If IID real $(X_n)$ have $S_n\mathrm{/}n$ converging almost surely to a finite, possibly random, limit $L$, then $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$ and $L=\mathbb{E}{X}_1$ almost surely.

Independent random elements have product joint law: Let $n\ge 1$, and let $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$ for $i<n$ be independent random elements. Define

$X=(X_0,\dots ,X_{n-1}):\mathrm{\Omega }\to {\prod }_{i<n}{S}_i\mathrm{.}$

Then $X$ is a random element of $({\prod }_{i<n}{S}_i,{⨂}_{i<n}{\mathrm{\Sigma }}_i)$, and its law is the finite product of the marginal laws:

${\mathbb{P}}_X={⨂}_{i<n}{\mathbb{P}}_{X_i}\mathrm{.}$

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be $\sigma$-finite measure spaces, and let $f:X\times Y\to [0,\mathrm{\infty }]$ be product-measurable. Then $x\mapsto {\int }_Y{f}_{x}d\nu$ and $y\mapsto {\int }_X{f}^{y}d\mu$ are measurable, and

${\int }_{X\times Y}fd(\mu \times \nu )={\int }_X\left({\int }_Y{f}_{x}d\nu \right)d\mu ={\int }_Y\left({\int }_X{f}^{y}d\mu \right)d\nu \mathrm{.}$

The principal inverse tangent $\arctan :\mathbb{R}\to (-\pi \mathrm{/}2,\pi \mathrm{/}2)$: By lem-tangent-principal-branch-is-bijective, tangent restricts to a continuous strictly increasing bijection

$\tan :(-\pi \mathrm{/}2,\pi \mathrm{/}2)⟶\mathbb{R}\mathrm{.}$

Its inverse is the principal inverse tangent

$\arctan :\mathbb{R}⟶(-\pi \mathrm{/}2,\pi \mathrm{/}2)\mathrm{.}$

Thus $\tan (\arctan y)=y$ for every real $y$, while $\arctan (\tan x)=x$ precisely for $x$ in the displayed principal interval. The inverse is continuous and strictly increasing by thm-continuous-inverse.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).