Measurable spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 18 March 2011

Measurable spaces, measurable sets, and measurable functions

A measurable space is a set $X$ with a specification of the measurable subsets of $X$ where it is required that

(a) $X$ is measurable,

(b) complements of measurable sets are measurable,

(c) countable unions of measurable sets are measurable,

(d) countable intersections of measurable sets are measurable,

More precisely, let $X$ be a set. A $\sigma$-algebra on $X$ is a set $ℳ$ of subsets of $X$ such that

(a) $X\in ℳ$,

(b) If $A\in ℳ$ then $A^c\in ℳ$,

(c) If $A_1,A_2,\dots \in ℳ$ then $A_1\cup A_2\cup \dots \in ℳ$

(d) If $A_1,A_2,\dots \in ℳ$ then $A_1\cap A_2\cap \dots \in ℳ$

HW: Show that (d) is redundant.

A measurable space is a set $X$ with a $\sigma$-algebra $ℳ$ on $X$.

Let $(X,ℳ)$ be a measurable space. A measurable set is a set in $ℳ$.

Measurable functions

Let $(X,ℳ)$ be a measurable space. A simple measurable function is an element of

$\mathrm{span-}\left\{{\chi }_A\right|A\in ℳ\}$,      where ${\chi }_A\left(x\right)=\{\begin{array}{cc}1,& \text{if}x\in A,\\ 0,& \text{otherwise,}\end{array}$

is the characteristic function of $A$.

Let $Y$ be a topological space. A measurable function from $X$ to $Y$ is a function $f:X\to Y$ such that

if $V$ is open in $Y$      then $f^{-1}\left(V\right)$ is measurable in $X$.

Borel measurability

Let $X$ be a topological space with topology $𝒯$. A Borel set is a subset $A$ of $X$ such that $A\in ℬ$, where $ℬ$ is the $\sigma$-algebra on $X$ generated by $𝒯$.

Let $X$ be a locally compact Hausdorff topological space. A Borel measure on $X$ is a measure $\mu :ℬ\to [0,\infty ]$, where $ℬ$ is the $\sigma$-algebra of Borel sets on $X$.

Notes and References

These notes were written for a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, §1.3].

References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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