1. ## finite measure space

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.

2. 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.