finite measure space

**eskimo343** · November 17th 2009, 08:21 AM

Let $(X, \mathcal{B}, \mu)$ be a finite measure space and $f$ be a nonnegative measurable function of $X$ . For each $A \in \mathcal{B}$ , set
$\nu(A)=\int_Af d\mu$ .

Verify that it is a finite measure if and only if $f$ is integrable.

I do not see how this if and only if follows. Any hints on this would be great. Thanks in advance.

**Focus** · November 17th 2009, 04:06 PM

Originally Posted by eskimo343

Let $(X, \mathcal{B}, \mu)$ be a finite measure space and $f$ be a nonnegative measurable function of $X$ . For each $A \in \mathcal{B}$ , set
$\nu(A)=\int_Af d\mu$ .

Verify that it is a finite measure if and only if $f$ is integrable.

I do not see how this if and only if follows. Any hints on this would be great. Thanks in advance.

- Note that $\nu(X)=||f||$ . This is the main connection with finiteness and integrability.

- Obviously $\nu(\emptyset)=0$ . For the linearity, first prove it for finite union (using the fact that $\mathbf{1}_{A\cup B}=\mathbf{1}_{A}+\mathbf{1}_{B}-=\mathbf{1}_{A\cap B}$ , then use dominated convergence to finish the result.

Let me know if you need any more help.

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