I was doing a problem given in ISI exam.It is a problem on measure theory.The problem is as follows:
Let $(\Omega,\mathcal F,\mu)$ be a measure space and $(E_n)$ be a sequence in $\mathcal F$ such that $\sum\limits_{n=1}^\infty \mu(E_n)<\infty$,then show that the set $A=\{\omega\in \Omega: \omega\in E_n$ for infinitely many $n\}$ is measurable and has measure $0$.
I solved the problem as follows:
Notice that,$A=\bigcap_{k=1}^\infty\bigcup_{n=k}^\infty E_n$ which is measurable because countable union and intersection of measurable sets is measurable.Again observe that $A\subset \bigcap_{k=1}^N\bigcup_{n=k}^\infty E_n=\bigcup\limits_{n=N}^\infty E_n$ so that $\mu(A)\leq \sum\limits_{n=N}^\infty \mu(E_n)\to 0$ as $N\to \infty$ because $\sum\limits_{n=1}^\infty \mu(E_n)<\infty$.Is my approach correct?