Hi!

How would one show the following:

1. If $\displaystyle \mu(X)<\infty$, $\displaystyle \mu$ is a measure iff $\displaystyle \mu$ is continuous from above (i.e. if $\displaystyle \{E_j\}_1^\infty \subset M\subset P(X), E_1\supset E_2\supset \ldots$ and $\displaystyle \mu(E_1)<\infty$ then $\displaystyle \mu(\cub_1^{\infty}E_j)=lim_{j\rightarrow\infty}(E _j)$).

($\displaystyle \Rightarrow$) I managed this direction, it's the ($\displaystyle \Leftarrow$) that I need help with.

2. If $\displaystyle \mu$ is semifinite and $\displaystyle \mu(E)=\infty$, for $\displaystyle C>0$ there exists $\displaystyle F\subset E$ with $\displaystyle C<\mu(F)<\infty$.

Ok, so by def of semifinite we know that ther exists an $\displaystyle F'\subset E$ such that $\displaystyle 0<\mu(F')<\infty$. Could I just set $\displaystyle F=F'$?