Real Analysis Folland, Problem 1.3.8 Measures

If $(X,M,\mu)$ is a measure space and $\{E_j\}_{1}^{\infty}\subset M$, then $\mu(\liminf E_j) \leq \liminf \mu(E_j)$.

Attempted proof - Let $(X,M,\mu)$ be a measure space and $\{E_j\}_{1}^{\infty}\subset M$. Notice that $\bigcap_{j=k}^{\infty} E_j = \cap E_k\cap E_{k+1}\cap\ldots$ which is increasing so $\{\bigcap_{1}^{\infty}E_k\}$ is an increasing sequence of sets so $\mu(\bigcap_{j=k}^{\infty}E_j) \leq \mu(E_k)$. We know that $$\liminf E_j = \bigcup_{k=1}^{\infty}\bigcap_{j=k}^{\infty}E_j$$ So $$\mu(\liminf E_j) = \mu\left(\bigcup_{k=1}^{\infty}\left(\bigcap_{j=k}^{\infty}E_j\right)\right) = \lim_{k\rightarrow \infty}\mu\left(\bigcap_{j=k}^{\infty}E_j\right) = \liminf\mu\left(\bigcap_{j=k}^{\infty}E_j\right) \leq \liminf\mu(E_j)$$ Therefore we must have $$\mu(\liminf E_j) \leq \liminf \mu(E_j)$$

I am somewhat skeptical of this result if anyone could provide some reasoning I would greatly appreciate it.

• See here, continuity of measures
– Moya
May 14 '16 at 15:25
• I know that proposition but I do not understand how it equals $\liminf....$ May 14 '16 at 15:28
• $\mu(\bigcap_{j=k}^{\infty}E_j) \leq \mu(E_j)$ should be $\mu(\bigcap_{j=k}^{\infty}E_j) \leq \mu(E_k)$. The $j$ is a running index. May 14 '16 at 15:37

Let $F_{k}=\bigcap\limits_{j=k}^{\infty} E_{j}$, then $\{F_{k}\}$ is an increasing sequence of sets. By definition $\lim\inf E_{j}=\bigcup\limits_{k=1}^{\infty} F_{k}$.
By Theorem 1.8.c (Continuity from below) in Folland's book, we get $$\mu\left(\lim\inf E_{j}\right)=\mu\left(\bigcup\limits_{k=1}^{\infty} F_{k}\right)=\lim\limits_{k\rightarrow \infty}\mu\left(F_{k}\right)=\lim\limits_{k\rightarrow \infty}\mu\left(\bigcap\limits_{j=k}^{\infty} E_{j}\right)$$ By definition $$\lim\inf\mu\left(E_{j}\right)=\lim\limits_{k\rightarrow \infty}\inf\{\mu\left(E_{j}\right) \mid j\geqslant k\}$$ Since $\mu\left(\bigcap\limits_{j=k}^{\infty} E_{j}\right)\leqslant \mu\left(E_{j}\right)$ for all $j \geqslant k$, we must have $\mu\left(\bigcap\limits_{j=k}^{\infty} E_{j}\right)\leqslant \inf\{\mu\left(E_{j}\right) \mid j\geqslant k\}$
Therefore, we have $$\mu\left(\lim\inf E_{j}\right)\leqslant\lim\inf\mu\left(E_{j}\right)$$
• It makes sense to that since $\mu\left(\bigcap_{j=k}^{\infty}E_j\right)\leq \mu(E_j)$ then we should have the result but I am somewhat skeptical May 14 '16 at 16:52
• $\lim\limits_{k\rightarrow \infty}\mu\left(\bigcap_{j=k}^{\infty}E_j\right)\leqslant\mu(E_j)$ is kind of unclear to me. It seems like you view $j$ as a fixed index. I think you should use $E_k$ instead of $E_j$. May 15 '16 at 15:46
• $\lim\limits_{k\rightarrow \infty}\mu\left(\bigcap_{j=k}^{\infty}E_j\right)=\liminf\mu\left(\bigcap_{j=k}^{\infty}E_j\right)\leqslant \liminf \mu(E_{k})=\liminf \mu(E_{j})$ May 15 '16 at 15:49