Let $(\Bbb{X}, \Sigma, \mu)$ be a measure space, The property of upper semicontinuity means that if $A_n$ is a decreasing sequence of measurable sets, namely $A_1 \supset A_2 \supset \ ...$ , and $\mu(A_1) \lt \infty$ then $$ \mu(\bigcap_{n=1}^\infty A_n) = \lim_{n\to \infty}\mu(A_n) $$
My question is - why is the condition $\mu(A_1) \lt \infty$ necessary? Why can't it be $\infty$? Any hints?
Thanks.
Under the setting of Lebesgue measure space:
$A_{n}=[n,\infty)$, $\mu(A_{n})=\infty$, so $\lim_{n\rightarrow\infty}\mu(A_{n})=\infty$ but $\displaystyle\bigcap_{n}A_{n}=\emptyset$, so $\mu\left(\displaystyle\bigcap_{n}A_{n}\right)=0$.