Standard Cauchy probability law

Given: Lebesgue measure $\lambda$ on $\mathbb{R}$ and the function $f(x)=1\mathrm{/}(\pi (1+x^2))$.

Assume the Axiom of Countable Choice, inherited by the Borel-to-Lebesgue and continuous-function measurability interfaces. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

On $\mathbb{R}$, Lebesgue measure is the restriction of Lebesgue outer measure to the Lebesgue sigma-algebra. Under Countable Choice this is a complete measure space, and every Borel set is Lebesgue measurable. Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$ Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable

The function $f$ is continuous because polynomials, constants, scalar multiples, and quotients with nonzero denominator preserve continuity; a continuous function $\mathbb{R}\to \mathbb{R}$ is Borel measurable under Countable Choice. Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function Continuous functions on Euclidean spaces are Borel measurable

A nonnegative Lebesgue-measurable density $f$ defines the measure $fd\lambda (A)={\int }_{A}fd\lambda$. The measure with density $f$ relative to $\mu$

A continuous function on a closed bounded interval is bounded and Riemann integrable; under Countable Choice its Lebesgue integral there equals its Riemann integral. A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral

$\arctan x={\int }_0^x(1+t^2{)}^{-1}dt$ for every real $x$, and oriented integrals are additive over adjacent intervals. Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$

The principal inverse tangent is a strictly increasing bijection from $\mathbb{R}$ onto $(-\pi \mathrm{/}2,\pi \mathrm{/}2)$. The principal inverse tangent $\arctan :\mathbb{R}\to (-\pi \mathrm{/}2,\pi \mathrm{/}2)$

Tangent is odd on its domain, so its inverse on the symmetric principal interval is odd. Pythagorean and parity identities for all six trigonometric functions on their natural domains

If nonnegative measurable functions $g_n$ increase pointwise to $g$, then $\int g_{n}d\lambda \uparrow \int gd\lambda$. Monotone convergence for the integral

A measure is a probability measure when its total mass is one. Probability measures and probability spaces Also, $\pi >0$. Pi as twice the smallest positive zero of cosine

By The real numbers and The Cauchy-sequence reals have the least-upper-bound property, the real numbers are a complete ordered field. Thus Every complete ordered field is Archimedean gives, for every real $r$, a natural $n$ with $n>r$.

If $E$ is measurable and $g$ is nonnegative measurable, then $g{\mathbf{1}}_E$ is measurable. Closure properties of measurable functions used by the integral