## 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

$\mathcal{M}$
of subsets of

$X$ such that

- (a)
$X\in \mathcal{M}$,
- (b)
If $A\in \mathcal{M}$
then
${A}^{c}\in \mathcal{M}$,
- (c)
If ${A}_{1},{A}_{2},\dots \in \mathcal{M}$
then
${A}_{1}\cup {A}_{2}\cup \dots \in \mathcal{M}$
- (d)
If ${A}_{1},{A}_{2},\dots \in \mathcal{M}$
then
${A}_{1}\cap {A}_{2}\cap \dots \in \mathcal{M}$

HW: Show that (d) is redundant.

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

Let
$(X,\mathcal{M})$ be a measurable space.
A **measurable set** is a set in $\mathcal{M}$.

## Measurable functions

Let $(X,\mathcal{M})$ be a measurable space.
A **simple measurable function** is an element of

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

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 $\mathcal{T}$. A **Borel set**
is a subset $A$ of $X$
such that $A\in \mathcal{B}$,
where $\mathcal{B}$ is the $\sigma $-algebra
on $X$ generated by $\mathcal{T}$.

Let $X$ be a locally compact Hausdorff topological
space. A **Borel measure** on $X$ is
a measure
$\mu :\mathcal{B}\to [0,\infty ]$,
where $\mathcal{B}$ 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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