Measure space

$\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 $\A$.

The product of two (or finitely many) measure spaces is a well-defined measure space.

A probability space is a measure space $(X,\A,\mu)$ satisfying $\mu(X)=1$. The product of infinitely many probability spaces is a well-defined probability space. (See [D, Sect. 8.2], [B, Sect. 3.5], [P].)

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 $\forall n \;\; \mu(A_n)<\infty$. (Finite measures are also σ-finite.)

Let $\mu(X)<\infty$. Both $(X,\A,\mu)$ and $\mu$ are called perfect if for every $\mu$-measurable (or equivalently, for every $\A$-measurable) function $f:X\to\R$ the image $f(X)$ contains a Borel (or equivalently, σ-compact) subset $B$ whose preimage $f^{-1}(B)$ is of full measure. (See [B, Sect. 7.5].)

For standard probability spaces see the separate article. Standard measure spaces are defined similarly. They are perfect, and admit a complete classification (unlike perfect measure spaces in general).

Examples. The real line with Lebesgue measure on Borel σ-algebra is an incomplete σ-finite measure space. The real line with Lebesgue measure on Lebesgue σ-algebra is a complete σ-finite measure space. The unit interval $(0,1)$ with Lebesgue measure on Lebesgue σ-algebra is a standard probability space. The product of countably many copies of this space is standard; for uncountably many factors the product is perfect but nonstandard. The one-dimensional Hausdorff measure on the plane is not σ-finite.

References

[T]	Terence Tao, "An introduction to measure theory", AMS (2011).   MR2827917   Zbl 05952932
[C]	Donald L. Cohn, "Measure theory", Birkhäuser (1993).   MR1454121   Zbl 0860.28001
[P]	David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).   MR1873379   Zbl 0992.60001
[B]	V.I. Bogachev, "Measure theory", Springer-Verlag (2007).   MR2267655  Zbl 1120.28001
[D]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001