Difference between revisions of "Measurable space"
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* 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). | * 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 $f_i:X\to Y_i$. | + | 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 $f_i:X\to Y_i$. On the other hand, if $Y$ is $\R$ (or $\C$, $\R^n$ etc.) then $\B$ is by default the Borel σ-algebra. |
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 $p_i:X\to X_i$. | 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 $p_i:X\to X_i$. | ||
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| + | Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a topological space is, by definition, generated by the open sets. | ||
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| + | Example. The following three σ-algebras on a separable Hilbert space $H$ are equal: | ||
| + | * the σ-algebra generated by the linear functionals $ x \mapsto \langle x,y \rangle $ for $y\in H$; | ||
| + | * the Borel σ-algebra corresponding to the norm topology on $H$; | ||
| + | * the Borel σ-algebra corresponding to the weak topology on $H$. | ||
| + | That is instructive: topological spaces are not a prerequisite to measurable spaces. | ||
====Older terminology==== | ====Older terminology==== | ||
Revision as of 12:12, 21 December 2011
Borel space
$ \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \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 $f_i:X\to Y_i$. On the other hand, if $Y$ is $\R$ (or $\C$, $\R^n$ etc.) then $\B$ is by default the Borel σ-algebra.
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 $p_i:X\to X_i$.
Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a topological space is, by definition, generated by the open sets.
Example. The following three σ-algebras on a separable Hilbert space $H$ are equal:
- the σ-algebra generated by the linear functionals $ x \mapsto \langle x,y \rangle $ for $y\in H$;
- the Borel σ-algebra corresponding to the norm topology on $H$;
- the Borel σ-algebra corresponding to the weak topology on $H$.
That is instructive: topological spaces are not a prerequisite to measurable spaces.
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 |
Measurable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_space&oldid=19871