# A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary

## Theorem

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

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

Let $g: X \to \overline \R$ be $\Sigma$-measurable.

Suppose that $f = g$ almost everywhere.

Then $g$ is also $\mu$-integrable, and:

- $\ds \int f \rd \mu = \int g \rd \mu$

## Proof

From Function Measurable iff Positive and Negative Parts Measurable, we have that:

- $g^+$, $f^+$, $g^-$ and $f^-$ are all $\Sigma$-measurable.

From Functions A.E. Equal iff Positive and Negative Parts A.E. Equal, we have that:

- $f^+ = g^+$ and $f^- = g^-$ $\mu$-almost everywhere.

Since $f^+$ and $g^+$ are positive $\Sigma$-measurable functions that are equal $\mu$-almost everywhere, we have:

- $\ds \int f^+ \rd \mu = \int g^+ \rd \mu$

from A.E. Equal Positive Measurable Functions have Equal Integrals.

Similarly, we have that $f^-$ and $g^-$ are positive $\Sigma$-measurable functions that are equal $\mu$-almost everywhere, and so:

- $\ds \int f^- \rd \mu = \int g^- \rd \mu$

Since $f$ is $\mu$-integrable, we have that:

- $\ds \int f^+ \rd \mu < \infty$

and:

- $\ds \int f^- \rd \mu < \infty$

Hence:

- $\ds \int g^+ \rd \mu < \infty$

and:

- $\ds \int g^- \rd \mu < \infty$

So $g$ is $\mu$-integrable.

We also have:

\(\ds \int g \rd \mu\) | \(=\) | \(\ds \int g^+ \rd \mu - \int g^- \rd \mu\) | Definition of Integral of Integrable Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \int f^+ \rd \mu - \int f^- \rd \mu\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \int f \rd \mu\) | Definition of Integral of Integrable Function |

$\blacksquare$