# finite measure space

• Nov 17th 2009, 08:21 AM
eskimo343
finite measure space
Let $\displaystyle (X, \mathcal{B}, \mu)$ be a finite measure space and $\displaystyle f$ be a nonnegative measurable function of $\displaystyle X$. For each $\displaystyle A \in \mathcal{B}$, set
$\displaystyle \nu(A)=\int_Af d\mu$.

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

I do not see how this if and only if follows. Any hints on this would be great. Thanks in advance.
• Nov 17th 2009, 04:06 PM
Focus
Quote:

Originally Posted by eskimo343
Let $\displaystyle (X, \mathcal{B}, \mu)$ be a finite measure space and $\displaystyle f$ be a nonnegative measurable function of $\displaystyle X$. For each $\displaystyle A \in \mathcal{B}$, set
$\displaystyle \nu(A)=\int_Af d\mu$.

Verify that it is a finite measure if and only if $\displaystyle f$ 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 $\displaystyle \nu(X)=||f||$. This is the main connection with finiteness and integrability.

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

Let me know if you need any more help.