Strong Laws of Large Numbers

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 $n\ge 2$).

Identical distribution and IID families: Let $(X_i{)}_{i\in I}$ be random elements with the same measurable target $(E,\mathcal{E})$. They are identically distributed if $\mathbb{P}(X_i\in B)=\mathbb{P}(X_j\in B)$ for all $i,j\in I$ and $B\in \mathcal{E}$, that is, their laws in def-law-or-distribution-of-a-random-element agree. They are independent and identically distributed (IID) if, in addition, the whole family is independent in def-independent-random-elements. Independence means mutual independence, not merely pairwise independence. No moment assumption is part of either definition. The empty family satisfies these universal conditions vacuously.

Kolmogorov strong law for independent uniformly bounded variances: Independent square-integrable real $(X_n{)}_{n\ge 1}$ with $C={\sup }_n\mathrm{Var}(X_n)<\mathrm{\infty }$ satisfy $(S_n-\mathbb{E}{S}_n)\mathrm{/}n\to 0$ almost surely.

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 $\int f_{n}d\mu \uparrow \int fd\mu \mathrm{.}$

Tail sum integrability equivalence: For a measurable $X:\mathrm{\Omega }\to [0,\mathrm{\infty }]$ on a probability space, ${\sum }_{n\ge 1}\mathbb{P}(X>n)\le \mathbb{E}X\le 1+{\sum }_{n\ge 1}\mathbb{P}(X>n)$. Thus $\mathbb{E}X<\mathrm{\infty }$ if and only if the tail series is finite.

First Borel-Cantelli lemma for events: Let $(A_n{)}_{n\in \mathbb{N}}$ be events in a probability space. If ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)<+\mathrm{\infty },$ then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=0.$

No independence hypothesis is needed.

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.

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: Identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$, along the embeddings $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$ (lem-nat-embeds-int, lem-int-embeds-rat, lem-rat-embeds-dense, def-integers). Then for every real $x$ there is exactly one integer $m$ with

$m\le x<m+1.$

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$.

Two independent ingredients are needed and neither may be dropped. Existence is the Archimedean property (thm-of-archimedean) together with the well-ordering of $\mathbb{N}$ (thm-well-ordering-principle): the first says that $x$ is caught between two integers at all, the second picks the least integer above $x$. Uniqueness is the discreteness of $\mathbb{Z}$: no integer lies strictly between $m$ and $m+1$.

This lemma is stated once here and reused. It is what turns "the nearest integer to $x$" from a picture into an object, and the companion page's oscillator $\psi (x)={\inf }_{n\in \mathbb{Z}}\mathrm{\mid }x-n\mathrm{\mid }$ is computed from it in one line.

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 $\int f_{n}d\mu \uparrow \int fd\mu \mathrm{.}$

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

If $x_k\to L$ then ${\sigma }_n\to L$: convergence implies $(C,1)$-summability to the same value: Let $(x_k)$ be a sequence of reals that converges (def-sequence, def-real-limit), and let $({\sigma }_n)$ be its sequence of Cesaro means (def-cesaro-mean). Then $({\sigma }_n)$ converges as well, and

${\lim }_n{\sigma }_n={\lim }_k{x}_k\mathrm{.}$

Both limits are asserted to exist: the right-hand one by hypothesis, the left-hand one as part of the conclusion. Equivalently: a convergent sequence is $(C,1)$-summable, to its own limit. The notation is licensed by uniqueness of limits of real sequences (lem-limit-unique).

The converse is false (fs-cesaro-converse).

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.

Summability of truncated normalized variances: For identically distributed integrable real $(X_n)$ and $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$, ${\sum }_{n\ge 1}\mathrm{Var}(Y_n)\mathrm{/}{n}^2\le 2\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$. No independence is required.

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 $n\ge 2$).

Cesaro limit of truncated means: For identically distributed integrable real $(X_n)$, with $Y_k=X_k{\mathbf{1}}_{\{\mathrm{\mid }{X}_k\mathrm{\mid }\le k\}}$, one has $n^{-1}{\sum }_{k=1}^n\mathbb{E}{Y}_k\to \mathbb{E}{X}_1$.

Iid linear truncation occurs only finitely often: For identically distributed integrable real $(X_n)$, put $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$. Almost surely $Y_n=X_n$ for all sufficiently large $n$. Consequently $n^{-1}{\sum }_{k=1}^n(X_k-Y_k)\to 0$. Independence is unnecessary.

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 $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=1.$

Tail sum integrability equivalence: For a measurable $X:\mathrm{\Omega }\to [0,\mathrm{\infty }]$ on a probability space, ${\sum }_{n\ge 1}\mathbb{P}(X>n)\le \mathbb{E}X\le 1+{\sum }_{n\ge 1}\mathbb{P}(X>n)$. Thus $\mathbb{E}X<\mathrm{\infty }$ if and only if the tail series is finite.

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.

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.

Independence forces covariance to vanish: If $X$ and $Y$ are independent square-integrable real random variables, then $\mathrm{Cov}(X,Y)=0.$

Thus independence implies zero covariance. The converse is false in general.

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.

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: Identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$, along the embeddings $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$ (lem-nat-embeds-int, lem-int-embeds-rat, lem-rat-embeds-dense, def-integers). Then for every real $x$ there is exactly one integer $m$ with

$m\le x<m+1.$

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$.

Two independent ingredients are needed and neither may be dropped. Existence is the Archimedean property (thm-of-archimedean) together with the well-ordering of $\mathbb{N}$ (thm-well-ordering-principle): the first says that $x$ is caught between two integers at all, the second picks the least integer above $x$. Uniqueness is the discreteness of $\mathbb{Z}$: no integer lies strictly between $m$ and $m+1$.

This lemma is stated once here and reused. It is what turns "the nearest integer to $x$" from a picture into an object, and the companion page's oscillator $\psi (x)={\inf }_{n\in \mathbb{Z}}\mathrm{\mid }x-n\mathrm{\mid }$ is computed from it in one line.

Integer powers $a^m$: Let $a\in \mathbb{R}$, where $\mathbb{R}$ is the ambient ordered field (def-ordered-field, def-field).

Natural exponents. By the recursion theorem (thm-recursion) applied to the set $\mathbb{R}$, the starting element $1$ and the function $f(x)=x\cdot a$, there is a unique function $\mathbb{N}\to \mathbb{R}$, written $n\mapsto a^n$, with

$a^0=1,a^{n+1}=a^n\cdot a(n\in \mathbb{N})\mathrm{.}$

Thus $a^1=a$, $a^2=a\cdot a$, and so on. Note that this is defined for every $a$, including $a=0$.

Negative exponents. If $a\mathrm{\ne }0$ and $n\in \mathbb{N}$ with $n\ge 1$, set

$a^{-n}:=(a^n{)}^{-1}\mathrm{.}$

Why that is legitimate. The right-hand side presupposes that $a^n$ is invertible, that is, that $a^n\mathrm{\ne }0$. This is a proof obligation and not an observation, and it is discharged by claim 2 of lem-power-laws: for $a\mathrm{\ne }0$ in a field, $a^n\mathrm{\ne }0$ for every $n\in \mathbb{N}$, proved there by induction on $n$ from the fact that a field has no zero divisors (lem-of-no-zero-divisors). That lemma is a statement about the operation introduced here, so it depends on this definition and is recorded in this item's justified_by rather than in its deps (SCHEMA §3). Given $a^n\mathrm{\ne }0$, the value $(a^n{)}^{-1}$ is a single well-determined element, because multiplicative inverses in a field are unique (lem-of-inverse-unique).

Integer exponents. Every integer $m$ (def-integers) is either $\iota (n)$ or $-\iota (n)$ for a unique natural $n$, where $\iota$ is the embedding $\mathbb{N}\to \mathbb{Z}$ (lem-nat-embeds-int, def-int-operations). This too is a citation and not a slogan: the order on $\mathbb{Z}$ is total (thm-int-ordered-ring), so $m\ge 0$ or $m<0$; the image of $\iota$ is exactly the set of nonnegative integers, and each of them is $\iota (n)$ for a unique natural $n$ (lem-nat-embeds-int); and if $m<0$ then $-m>0$, by compatibility of the order with addition (thm-int-ordered-ring), so $-m=\iota (n)$ and $m=-\iota (n)$, with $n$ unique because $\iota$ is injective. The two clauses above therefore define $a^m$ for every $m\in \mathbb{Z}$ whenever $a\mathrm{\ne }0$, and for every $m\in \mathbb{N}$ for arbitrary $a$. The clauses are consistent where they overlap: the only overlap is $m=0$, where $-\iota (0)=\iota (0)$ and $(a^0{)}^{-1}=1^{-1}=1=a^0$.

For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges: Let $r\in \mathbb{R}$ and let $r^k$ be the integer power (def-integer-power), so that $r^0=1$ for every $r$, including $r=0$.

1. If $\mathrm{\mid }r\mathrm{\mid }<1$ then the series $\sum r^k$ converges (def-series) and ${\sum }_{k=0}^{\mathrm{\infty }}{r}^k=\frac{1}{1-r}\mathrm{.}$
2. If $\mathrm{\mid }r\mathrm{\mid }\ge 1$ then $\sum r^k$ diverges.

The series starts at $k=0$ and its first term is $r^0=1$; in particular ${\sum }_{k=0}^{\mathrm{\infty }}{2}^{-k}=2$, while the series starting at $k=1$ sums to $1$. Which starting index is meant has to be said, and it is said here.

Summability of truncated normalized variances: For identically distributed integrable real $(X_n)$ and $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$, ${\sum }_{n\ge 1}\mathrm{Var}(Y_n)\mathrm{/}{n}^2\le 2\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$. No independence is required.

Chebyshev's inequality for random variables: If $X$ is a square-integrable real random variable and $a>0$, then $\mathbb{P}(\mathrm{\mid }X-\mathbb{E}[X]\mathrm{\mid }\ge a)\le \frac{\mathrm{Var}(X)}{a^2}\mathrm{.}$

First Borel-Cantelli lemma for events: Let $(A_n{)}_{n\in \mathbb{N}}$ be events in a probability space. If ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)<+\mathrm{\infty },$ then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=0.$

No independence hypothesis is needed.

Cesaro limit of truncated means: For identically distributed integrable real $(X_n)$, with $Y_k=X_k{\mathbf{1}}_{\{\mathrm{\mid }{X}_k\mathrm{\mid }\le k\}}$, one has $n^{-1}{\sum }_{k=1}^n\mathbb{E}{Y}_k\to \mathbb{E}{X}_1$.

Iid linear truncation occurs only finitely often: For identically distributed integrable real $(X_n)$, put $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$. Almost surely $Y_n=X_n$ for all sufficiently large $n$. Consequently $n^{-1}{\sum }_{k=1}^n(X_k-Y_k)\to 0$. Independence is unnecessary.

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.

Almost-sure convergence implies convergence in probability: If $X_n\to X$ almost surely, then $X_n\to X$ in probability.

Measure-preserving transformations and systems: Let $(X,\mathcal{A},\mu )$ be a measure space. A measurable self-map $T:X\to X$ is measure preserving if $\mu (T^{-1}E)=\mu (E)$ for every $E\in \mathcal{A}$. The quadruple $(X,\mathcal{A},\mu ,T)$ is a measure-preserving system; it is a probability system if $\mu (X)=1$. Here $T^{-1}E=\{x:T(x)\in E\}$ denotes an inverse image, whether or not $T$ is invertible. Neither completeness nor finiteness is implicit. The measure-space and measurable-map conventions are def-measure-space and def-measurable-function-between-measurable-spaces.

Arithmetic and lattice operations preserve measurability whenever they are defined: Let $(X,\mathcal{A})$ be a measurable space and let $f,g:X\to \bar{\mathbb{R}}$ be measurable. Then:

1. $cf$ is measurable for every real scalar $c$;
2. $\max (f,g)$, $\min (f,g)$, $\mathrm{\mid }f\mathrm{\mid }$, $f^{+}$, and $f^{-}$ are measurable;
3. if $f+g$ is pointwise defined, then $f+g$ is measurable;
4. with the convention of rem-zero-times-infinity-convention-for-pointwise-products, the pointwise product $fg$ is measurable.

Integral invariance under measure-preserving maps: If $T$ preserves $\mu$ and $f:X\to [0,\mathrm{\infty }]$ is measurable, then $\int f\circ Td\mu =\int fd\mu$, allowing infinity. If $f$ is integrable real or complex valued, $f\circ T$ is integrable and the same equality holds. Conversely, for a measurable self-map, equality for every measurable indicator implies measure preservation.

The Lebesgue integral is linear on $L^1(\mu )$: The class $L^1(\mu )$ is a complex vector space, and the Lebesgue integral is complex-linear on it: $\int (\alpha f+\beta g)d\mu =\alpha \int fd\mu +\beta \int gd\mu (\alpha ,\beta \in \mathbb{C},f,g\in L^1(\mu ))\mathrm{.}$

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

The Lebesgue integral is linear on $L^1(\mu )$: The class $L^1(\mu )$ is a complex vector space, and the Lebesgue integral is complex-linear on it: $\int (\alpha f+\beta g)d\mu =\alpha \int fd\mu +\beta \int gd\mu (\alpha ,\beta \in \mathbb{C},f,g\in L^1(\mu ))\mathrm{.}$

Arithmetic and lattice operations preserve measurability whenever they are defined: Let $(X,\mathcal{A})$ be a measurable space and let $f,g:X\to \bar{\mathbb{R}}$ be measurable. Then:

1. $cf$ is measurable for every real scalar $c$;
2. $\max (f,g)$, $\min (f,g)$, $\mathrm{\mid }f\mathrm{\mid }$, $f^{+}$, and $f^{-}$ are measurable;
3. if $f+g$ is pointwise defined, then $f+g$ is measurable;
4. with the convention of rem-zero-times-infinity-convention-for-pointwise-products, the pointwise product $fg$ is measurable.

Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable: Let $(X,\mathcal{A})$ be a measurable space and let $f_n:X\to \bar{\mathbb{R}}$ be measurable for every $n\in \mathbb{N}$. Then the functions

${\sup }_n{f}_n,{\inf }_n{f}_n,{\mathrm{lim\hspace{0.17em}sup}}_n{f}_n,{\mathrm{lim\hspace{0.17em}inf}}_n{f}_n$

are measurable. The set

$\{x:{\lim }_n{f}_n(x)\text{ exists in }\bar{\mathbb{R}}\}$

is measurable. In particular, if $f_n\to f$ pointwise, then $f$ is measurable.

The maximal ergodic inequality on a probability space: Let $T$ preserve a probability measure $P$, and let $f$ be integrable, real-valued and measurable. Put $S_{k}f={\sum }_{j=0}^{k-1}f\circ T^j$, $M_N=\max (0,S_{1}f,\dots ,S_{N}f)$ and $E_N=\{M_N>0\}$ for $N\ge 1$. Then ${\int }_{E_N}fdP\ge 0$, and also ${\int }_{E}fdP\ge 0$ for $E=\{{\sup }_{k\ge 1}{S}_{k}f>0\}$.

Ergodicity relative to an invariant measure: A measure-preserving system is ergodic for $\mu$ if each $E\in \mathcal{I}$ has $\mu (E)=0$ or $\mu (X\setminus E)=0$, with $\mathcal{I}$ as in def-strict-and-mod-null-invariant-$\sigma$-algebras. For a probability system this means $\mu (E)\in \{0,1\}$. The definition is relative to the invariant measure; no probability assumption is implicit in the general null/conull formulation.

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 $0$.