Measure theory limit question 
Let $(X, \cal{M},\mu)$ be a finite positive measure space and $f$ a $\mu$-a.e. strictly positive measurable function on $X$.  If $E_n\in\mathcal{M}$, for $n=1,2,\ldots $ and $\displaystyle \lim_{n\rightarrow\infty} \int_{E_n}f d\mu=0$, prove that $\displaystyle\lim_{n\rightarrow\infty}\mu(E_n)=0$. 


Hints: 


*

*If $f$ is strictly positive almost everywhere then for "most of $X$" (which you can make arbitrarily large) there is a positive value it is greater than.  

*So unless $E_n$ gets "small enough" permanently in the tail, you can show $\int_{E_n}f d\mu$ most be greater than a particular positive value infinitely often, by comparing the overlap between $E_n$ and "most of $X$".

*You can formalise this, as with other limits, taking care with the relationship between "most" and "small enough".   
