Integral of Positive Measurable Function is Monotone/Corollary
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Corollary to Integral of Positive Measurable Function is Monotone
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g: X \to \overline \R$ be positive $\Sigma$-measurable functions.
Let $A \in \Sigma$.
Suppose that $f \le g$, where $\le$ denotes pointwise inequality.
Then:
- $\ds \int_A f \rd \mu \le \int_A g \rd \mu$
where the integral sign denotes $\mu$-integration over $A$.
This can be summarized by saying that $\ds \int_A \cdot \rd \mu$ is monotone.
Proof
From the definition of $\mu$-integration over $A$, we have:
- $\ds \int_A f \rd \mu = \int \paren {\chi_A \times f} \rd \mu$
and:
- $\ds \int_A g \rd \mu = \int \paren {\chi_A \times g} \rd \mu$
We show that:
- $f \times \chi_A \le g \times \chi_A$
If $x \in A$, we have:
\(\ds \map {\paren {f \times \chi_A} } x\) | \(=\) | \(\ds \map f x \map {\chi_A} x\) | Definition of Pointwise Multiplication of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x\) | Definition of Characteristic Function of Set |
and:
\(\ds \map {\paren {g \times \chi_A} } x\) | \(=\) | \(\ds \map g x \map {\chi_A} x\) | Definition of Pointwise Multiplication of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map g x\) | Definition of Characteristic Function of Set |
So:
- $\map {\paren {f \times \chi_A} } x \le \map {\paren {g \times \chi_A} } x$ for all $x \in A$.
Now take $x \in X \setminus A$.
We have:
\(\ds \map {\paren {f \times \chi_A} } x\) | \(=\) | \(\ds \map f x \map {\chi_A} x\) | Definition of Pointwise Multiplication of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of Characteristic Function of Set |
and:
\(\ds \map {\paren {g \times \chi_A} } x\) | \(=\) | \(\ds \map g x \map {\chi_A} x\) | Definition of Pointwise Multiplication of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Definition of Characteristic Function of Set |
So:
- $\map {\paren {f \times \chi_A} } x \le \map {\paren {g \times \chi_A} } x$ for all $x \in X \setminus A$
giving:
- $f \times \chi_A \le g \times \chi_A$
From Integral of Positive Measurable Function is Monotone, we therefore have:
- $\ds \int \paren {f \times \chi_A} \rd \mu \le \int \paren {g \times \chi_A} \rd \mu$
From the definition of $\mu$-integration over $A$, we have:
- $\ds \int_A f \rd \mu \le \int_A g \rd \mu$
$\blacksquare$