User:Boris Tsirelson/sandbox2

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A measure space is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a σ-algebra of its subsets, and $\mu:\A\to[0,+\infty]$ a measure. Thus, a measure space consists of a measurable space and a measure. The notation $(X,\A,\mu)$ is often shortened to $(X,\mu)$ and one says that $\mu$ is a measure on $X$; sometimes the notation is shortened to $X$.

Basic notions and constructions

Inner measure $\mu_*$ and outer measure $\mu^*$ are defined for all subsets $A\subset X$ by

$ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$

$A$ is called a null (or negligible) set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of full measure, and one says that $x\notin A$ for almost all $x$ (in other words, almost everywhere). Two sets $A,B\subset X$ are almost equal (or equal mod 0) if $(x\in A)\iff(x\in B)$ for almost all $x$ (in other words, $A\setminus B$ and $B\setminus A$ are negligible). Two functions $f,g:X\to Y$ are almost equal (or equal mod 0, or equivalent) if they are equal almost everywhere.

A subset $A\subset X$ is called measurable (or $\mu$-measurable) if it is almost equal to some $B\in\A$. In this case $\mu_*(A)=\mu^*(A)=\mu(B)$. If $\mu_*(A)=\mu^*(A)<\infty$ then $A$ is $\mu$-measurable. All $\mu$-measurable sets are a σ-algebra $\A_\mu$ containing $\A$.

Some classes of measure spaces

Let $(X,\A,\mu)$ be a measure space.

Both $(X,\A,\mu)$ and $\mu$ are called complete if $\A_\mu=\A$ or, equivalently, if $\A$ contains all null sets. The completion of $(X,\A,\mu)$ is the complete measure space $(X,\A_\mu,\tilde\mu)$ where $\tilde\mu(A)=\mu(B)$ whenever $A\in\A_\mu$ is almost equal to $B\in\A$.

If $X$ is a set of finite measure, that is, $\mu(X)<\infty$, then $\mu$, and sometimes also $(X,\A,\mu)$, is called finite.

Both $(X,\A,\mu)$ and $\mu$ are called σ-finite if $X$ can be split into countably many sets of finite measure, that is, $X=A_1\cup A_2\cup\dots$ for some $A_n\in\A$ such that $\mu(A_n)<\infty$. (Finite measures are also σ-finite.)