A.E. Equal Positive Measurable Functions have Equal Integrals
Corollary on its own page
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Contents
Theorem
Let $\struct {X, \Sigma, \mu}$ 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:
- $\displaystyle \int f \rd \mu = \int g \rd \mu$
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:
- $\displaystyle \int f \rd \mu = \int g \rd \mu$
Proof
Let $N$ be the set defined by:
- $N = \set {x \in X: \map f x \ne \map g x}$
By hypothesis, $N$ is a $\mu$-null set.
If $N = \O$, then $f = g$, trivially implying the result.
If $N \ne \O$, then by Set with Relative Complement forms Partition:
- $X = N \cup \paren {X \setminus N}$
Now:
| \(\displaystyle \int f \rd \mu\) | \(=\) | \(\displaystyle \int f \chi_X \rd \mu\) | Characteristic Function of Universe | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int f \chi_{N \cup \paren {X \setminus N} } \rd \mu\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int \map f {\chi_N + \chi_{X \setminus N} } \rd \mu\) | Characteristic Function of Union | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int f \chi_N \rd \mu + \int f \chi_{X \setminus N} \rd \mu\) | Integral of Integrable Function is Additive | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 0 + \int f \chi_{X \setminus N} \rd \mu\) | Integral of Integrable Function over Null Set | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int g \chi_{X \setminus N} \rd \mu\) | Definition of $N$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int g \chi_N \rd \mu + \int g \chi_{X \setminus N} \rd \mu\) | Integral of Integrable Function over Null Set | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int \map g {\chi_N + \chi_{X \setminus N} } \rd \mu\) | Integral of Integrable Function is Additive | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int g \chi_X \rd \mu\) | Characteristic Function of Union | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \int g \rd \mu\) | Characteristic Function of Universe |
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which establishes the result.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.10$
