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).

Let $X$ be a set, $(Y,\B)$ a measurable space, and $(f_i)_{i\in I}$ a family of maps $f_i:X\to Y$. The σ-algebra generated by these maps is defined as the smallest σ-algebra $\A$ on $X$ such that all $f_i$ are measurable from $(X,\A)$ to $(Y,\B)$. More generally, one may take measurable spaces $(Y_i,\B_i)$ and maps $f_i:X\to Y_i$.

Given a family of measurable spaces $(X_i,\A_i)$ for $i\in I$, their product is defined as the measurable space $(X,\A)$ where $X=\prod_i X_i$ is the direct product of sets, and $\A$ is generated by the projection maps $p_i:X\to X_i$.

Older terminology

Weaker assumptions on $\A$ were usual in the past. For example, according to [5], $\A$ need not contain the whole $X$, it is a σ-ring, not necessarily a σ-algebra. According to [6], 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]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) User:Boris Tsirelson/MR
[4]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) User:Boris Tsirelson/MR
[5]	Paul R. Halmos, "Measure theory", v. Nostrand (1950) User:Boris Tsirelson/MR
[6]	Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) User:Boris Tsirelson/MR