lim inf $\mu (E_j)$

Question

I'm trying to solve 1.3.8 from Folland's Real Analysis and I am to show that

$$\mu(\liminf E_j) \leq\liminf\mu (E_j)$$

where $\mu$ is a measure in some measure space $(X,M,\mu)$ and $\{ E_n \}_1^\infty \subseteq M$.

I don't want a proof or even a hint. I just don't understand the right side of the inequality. The definition of $\liminf E_n$ is $\bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n$ for some family of sets $\{ E_n \}$. The problem is that $\mu (E_j)$ is a number, not a set, so what dose it mean to say $\liminf\mu (E_j)$?

Answer 1

The right-hand side is the limit inferior of the sequence $\{\mu(E_j)\}_{j = 1}^\infty$, so it is $\lim\limits_{j\to \infty} [\inf\limits_{k \ge j} \mu(E_k)]$. More generally, if $\{a_n\}_{n = 1}^\infty$ is a sequence of real numbers, $\liminf a_n$ is defined as $\lim\limits_{n\to \infty} (\inf\limits_{k \ge n} a_k)$.