Integral of Positive Measurable Function is Positive Homogeneous/Corollary

Corollary to Integral of Positive Measurable Function is Positive Homogeneous

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.

Let $\lambda \in \overline \R$ be an extended real number with $\lambda \ge 0$.

Let $A \in \Sigma$.

Then:

$\ds \int_A \lambda f \rd \mu = \lambda \int_A f \rd \mu$

where:

$\lambda f$ is the pointwise $\lambda$-multiple of $f$

the integral sign denotes $\mu$-integration over $A$.

This can be summarized by saying that $\ds \int_A \cdot \rd \mu$ is positive homogeneous.

Proof

We have:

\(\ds \int_A \lambda f \rd \mu\)

\(=\)

\(\ds \int \paren {\lambda f \times \chi_A} \rd \mu\)

Definition of Integral of Positive Measurable Function over Measurable Set

\(\ds \)

\(=\)

\(\ds \lambda \int \paren {f \times \chi_A} \rd \mu\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds \lambda \int_A f \rd \mu\)

Definition of Integral of Positive Measurable Function over Measurable Set

$\blacksquare$