Integral of Positive Measurable Function is Positive Homogeneous

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

Let $f: X \to \R, f \in \MM_{\overline \R}^+$ be a positive measurable function.

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

Then:


 * $\ds \int \lambda f \rd \mu = \lambda \int f \rd \mu$

where:


 * $\lambda f$ is the pointwise $\lambda$-multiple of $f$
 * The integral sign denotes $\mu$-integration

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

Proof
Suppose that $\lambda < \infty$.

From Measurable Function is Pointwise Limit of Simple Functions, there exists an increasing sequence $\sequence {f_n}_{n \mathop \in \N}$ of positive simple functions such that:


 * $\ds \map f x = \lim_{n \mathop \to \infty} \map {f_n} x$

From Combination Theorem for Sequences: Real: Multiple Rule, we have:


 * $\ds \lambda \map f x = \lim_{n \mathop \to \infty} \paren {\lambda \map {f_n} x}$

for each $x \in X$.

We then have:

so we get the demand in the case $\lambda < \infty$.

Now suppose that $\lambda = \infty$.

We can write:


 * $\ds \lambda f = \lim_{k \mathop \to \infty} k f$

Since $f \ge 0$, the sequence $\sequence {k f}_{k \mathop \in \N}$ is increasing, we have:


 * $\ds \int \lambda f \rd \mu = \lim_{k \mathop \to \infty} \int k f \rd \mu$

from the monotone convergence theorem.

From our earlier work, we have:


 * $\ds \int k f \rd \mu = k \int f \rd \mu$

so that:

giving the demand in the case $\lambda = \infty$.

Also see

 * Integral of Positive Simple Function is Positive Homogeneous, a similar result for positive simple functions.