A.E. Equal Positive Measurable Functions have Equal Integrals

From ProofWiki
Jump to: navigation, search

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f, g: X \to \overline{\R}_{\ge 0}$ be positive $\mu$-measurable functions.

Suppose that $f = g$ almost everywhere.


Then:

$\displaystyle \int f \, \mathrm d \mu = \int g \, \mathrm d \mu$


Corollary

Let $f: X \to \overline{\R}$ be a $\mu$-integrable function, and $g: X \to \overline{\R}$ be measurable.

Suppose that $f = g$ almost everywhere.


Then $g$ is also $\mu$-integrable, and:

$\displaystyle \int f \, \mathrm d \mu = \int g \, \mathrm d \mu$


Proof

Let $N$ be the set defined by:

$N = \left\{{x \in X: f \left({x}\right) \ne g \left({x}\right)}\right\}$

By hypothesis, $N$ is a $\mu$-null set.

If $N = \varnothing$, then $f = g$, trivially implying the result.


If $N \ne \varnothing$, then by Set with Relative Complement forms Partition:

$X = N \cup \left({X \setminus N}\right)$

Now:

\(\displaystyle \int f \, \mathrm d \mu\) \(=\) \(\displaystyle \int f \chi_X \, \mathrm d \mu\) $\quad$ Characteristic Function of Universe $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int f \chi_{N \cup \left({X \setminus N}\right)} \, \mathrm d \mu\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int f \left({\chi_N + \chi_{X \setminus N} }\right) \, \mathrm d \mu\) $\quad$ Characteristic Function of Union $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int f \chi_N \, \mathrm d \mu + \int f \chi_{X \setminus N} \, \mathrm d \mu\) $\quad$ Integral of Integrable Function is Additive $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 + \int f \chi_{X \setminus N} \, \mathrm d \mu\) $\quad$ Integral of Integrable Function over Null Set $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int g \chi_{X \setminus N} \, \mathrm d \mu\) $\quad$ Definition of $N$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int g \chi_N \, \mathrm d \mu + \int g \chi_{X \setminus N} \, \mathrm d \mu\) $\quad$ Integral of Integrable Function over Null Set $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int g \left({\chi_N + \chi_{X \setminus N} }\right) \, \mathrm d \mu\) $\quad$ Integral of Integrable Function is Additive $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int g \chi_X \, \mathrm d \mu\) $\quad$ Characteristic Function of Union $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int g \, \mathrm d \mu\) $\quad$ Characteristic Function of Universe $\quad$

which establishes the result.

$\blacksquare$


Sources