A.E. Equal Positive Measurable Functions have Equal Integrals

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:


 * $\displaystyle \int f \rd \mu = \int g \rd \mu$