A.E. Equal Positive Measurable Functions have Equal Integrals
Corollary on its own page
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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 | $\quad$ |
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which establishes the result.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.10$
