Lebesgue Integral of Measurable Function

Definition

Consider a measure spaces $(\Omega ,\mathcal{F},\mu )$ and a Measure Space $(\overline{\mathbb{R}},\mathcal{R})$ and a Measurable Function $f:(\Omega ,\mathcal{F})\to (\overline{\mathbb{R}},\mathcal{R})$. Then the Lebesgue integral of the function $f$ with respect to $\mu$ is defined as ${\int }_{\Omega }fd\mu :={\int }_{\Omega }{f}^{+}d\mu -{\int }_{\Omega }{f}^{-}d\mu$ where $f^{+}:=\max (0,f)$ and $f^{-}:=\max (0,-f)$ for the function $f$.

Existence of Integral

$\int fd\mu$ exists

• $\int f^{+}d\mu <\infty$ and $\int f^{-}d\mu <\infty$ ($f$ is integrable w.r.t $\mu$) $\int fd\mu =\int f^{+}d\mu -\int f^{-}d\mu$
• $\int f^{+}d\mu =\infty$ and $\int f^{-}d\mu <\infty$ $\int fd\mu =\infty$
• $\int f^{+}d\mu <\infty$ and $\int f^{-}d\mu =\infty$ $\int fd\mu =-\infty$

$\int fd\mu$ not exists

• $\int f^{+}d\mu =\infty$ and $\int f^{-}d\mu =\infty$ $\int fd\mu$ is not defined

Notations

For a Density Function $F_X={\mu }_X((-\infty ,x])$ on the Measurable Space $(\mathbb{R},\mathcal{R},{\mu }_X)$ $\int gd{\mu }_X=\int gd{F}_X=\int g(x)d{F}_X(x)$

For a Density Function $F_X={\mu }_X((-\infty ,x])$ that has Density Function $f_X(x)$ on the Measurable Space $(\mathbb{R},\mathcal{R},{\mu }_X)$ $\int g(x)d{F}_X(x)=\int g(x)f_X(x)dx$

Consider a Measure Space $(\Omega ,\mathcal{F},\#)$ where $\Omega$ is a countable set, $\mathcal{F}=2^{\Omega }$ is a Sigma-Field defined on the set $\Omega$, and $\#$ is a Counting measure on $\mathcal{F}$. Then $\int fd\#=\sum _{i\in \Omega }f(i)$

Facts

For the functions $f^{+}:=\max (0,f)$ and $f^{-}:=\max (0,-f)$ of $f$,

• $f^{+},f^{-}$ are non-negative
• $|f|=f^{+}+f^{-}$
• $f=f^{+}-f^{-}$
• If $f$ is measurable, then $f^{+}$ and $f^{-}$ are also measurable.