## Measurable spaces

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 $\left(X,ℳ\right)$ be a measurable space. A measurable set is a set in $ℳ$.

## Measurable functions

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

 $\mathrm{span-}\left\{{\chi }_{A}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}A\in ℳ\right\}$,      where ${\chi }_{A}\left(x\right)=\left\{\begin{array}{cc}1,& \text{if}\phantom{\rule{1em}{0ex}}x\in A,\\ 0,& \text{otherwise,}\end{array}$
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 $\mu :ℬ\to \left[0,\infty \right]$, 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.