Measurable space

Borel space

$ \newcommand{\R}{\mathbb R} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\P}{\mathbf P} $ A set with a distinguished σ-algebra of subsets (called measurable). More formally: a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.

Examples: $\R^n$ with the Borel σ-algebra; $\R^n$ with the Lebesgue σ-algebra.

Let $(X,\A)$ and $(Y,\B)$ be measurable spaces.

• A map $f:X\to Y$ is called measurable if $f^{-1}(B) \in \A$ for every $B\in\B$.
• These two measurable spaces are called isomorphic if there exists a bijection $f:X\to Y$ such that $f$ and $f^{-1}$ are measurable (such $f$ is called an isomorphism).

Older terminology

Weaker assumptions on $\A$ were usual in the past. For example, according to [4], $\A$ need not contain the whole $X$, it is a σ-ring, not necessarily a σ-algebra. According to [5], a measurable space is not a pair $(X,\A)$ but a measure space $(X,\A,\mu)$ such that $X\in\A$ (and again, $\A$ is generally a σ-ring).

References

[1]	Terence Tao, "An introduction to measure theory", AMS (2011) User:Boris Tsirelson/MR
[2]	David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002) User:Boris Tsirelson/MR
[3]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) User:Boris Tsirelson/MR
[4]	Paul R. Halmos, "Measure theory", v. Nostrand (1950) User:Boris Tsirelson/MR
[5]	Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) User:Boris Tsirelson/MR