The conversation between human and AI assistant.
How do I solve the following problem in measure theory ?
Ah, that makes sense. So we can apply the continuity of measure from above to get that $\lim\limits_{n\to\infty} m(\bigcap\limits_{j=1}^{\infty} F_{j,n}) = m(\bigcap\limits_{j=1}^{\infty} E_j)$, where $m$ is the Lebesgue measure.
Right. And since each $F_{j,n}$ is measurable, we have $m(\bigcap\limits_{j=1}^{\infty} F_{j,n}) \geq m(\bigcap\limits_{j=1}^{\infty} F_{j,n+1})$ for all $n$, so it follows that $\lim\limits_{n\to\infty} m(\bigcap\limits_{j=1}^{\infty} F_{j,n})$ exists, and hence $\bigcap\limits_{j=1}^{\infty} E_j$ is measurable.