Measures and Integration

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

Last updates: 18 March 2011

Measures

Let $(X,ℳ)$ be a measurable space.

A positive measure on $(X,ℳ)$ is a function $\mu :ℳ\to [0,\infty ]$ such that

if $A_1,A_2,\dots \in ℳ$ are such that if $i,j\in {\mathbb{Z}}_{>0}$ and $i\ne j$ then $A_i\cap A_j=\varnothing$,

then ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mu \left(A_i\right)}$.

A complex measure on $(X,ℳ)$ is a function $\mu :ℳ\to ℂ$ such that

if $A_1,A_2,\dots \in ℳ$ are such that if $i,j\in {\mathbb{Z}}_{>0}$ and $i\ne j$ then $A_i\cap A_j=\varnothing$,

then ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mu \left(A_i\right)}$.

A measure on $(X,ℳ)$ is a positive measure or a complex measure on $(X,ℳ)$.

Integration with respect to positive measures

Let $(X,ℳ)$ be a measurable space. Let $\mu$ be a positive measure on $(X,ℳ)$ and let $E\in ℳ$. For a function $f:X\to [-\infty ,\infty ]$ let

$f^{+}=\frac{1}{2}\left(\right|f|+f)\text{and}$ f^{+}=\frac{1}{2}\left(\right|f|-f)$.$

For a simple measurable function

$s={\displaystyle \sum _{i=1}^n}{\alpha }_i{\chi }_{A_i}$      define      ${\displaystyle {\int }_{E}sd\mu =\sum _{i=1}^n}{\alpha }_i\mu (A_i\cap E)$.

For a measurable function $f:X\to [0,\infty ]$ define

${\displaystyle {\int }_{E}fd\mu }=\sup \left\{{\int }_{E}sd\mu \right|s\text{is simple measurable and}0\le s\le f\}$

For a measurable function $f:X\to [-\infty ,\infty ]$ such that ${\displaystyle {\int }_E{f}^{+}d\mu <\infty }$ or ${\displaystyle {\int }_E{f}^{-}d\mu <\infty }$ define

${\displaystyle {\int }_{E}fd\mu ={\int }_E{f}^{+}d\mu -{\int }_E{f}^{-}d\mu }$.

For a measurable function $f:X\to ℂ$ let $f=u+iv$ where $u:X\to ℝ$ and $v:X\to ℝ$ are measurable and define

${\displaystyle {\int }_{E}fd\mu ={\int }_{E}ud\mu +i{\int }_{E}vd\mu }$.

Integration with respect to complex measures

Let $(X,ℳ)$ be a measurable space. Let $\mu$ be a complex measure on $X$. Define a positive measure $\left|\mu \right|:ℳ\to [0,\infty ]$ by

${\displaystyle \left|\mu \right|\left(E\right)=\sup \left\{\sum _{i=1}^{\infty }\left|\mu \left(E_i\right)\right|\right|E_1,E_2,\dots \in ℳ\text{partition}E\}}$.

for a measurable function $f:X\to ℂ$ define

${\displaystyle {\int }_{X}fd\mu ={\int }_{X}fhd\left|\mu \right|}$,

where $h:X\to ℂ$ is a measurable function such that

if $x\in X$ then $\left|h\right(x)|=1$,     and     if $E\in ℳ$ then $\mu \left(E\right)={\displaystyle {\int }_{X}hd\left|\mu \right|}$.

HW: Use the Radon-Nikodym theorem to show that the function $h$ exists (see [Ru, Theorem 6.12]).

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, Chapters 1-6].

References

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

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