liminf inequality in measure spaces 
Let $(X;\mathscr{M},\mu)$ be a measure space and $\{E_j\}_{j=1}^\infty\subset \mathscr{M}$. Show that $$\mu(\liminf E_j)\leq \liminf \mu(E_j)$$
  and, if $\mu\left(\bigcup_{j=1}^\infty E_j\right)<\infty$, that
  $$\mu(\limsup E_j)\geq \limsup \mu(E_j).$$

I'm trying to parse what's going on.  On the left, we're taking the measure of $\liminf E_j$, which is $\cup_{i=1}^\infty\cap_{j=i}^\infty E_i$.  This is the union of the tails... okay.
On the right, we've got $\lim_{n\to\infty}\inf\{\mu(E_j):n\leq j\}$.  The smallest $\mu$ for everything after $n$ (or the greatest lower bound, anyway).
I can't make any progress, I've been stuck here for quite a while. I just don't know where to make the comparison. Can I get a nudge?

Apply Fatou's Lemma to the characteristic functions of these sets.
More precisely, use the notation $\chi_A$ to denote the characteristic function of the set $A$.
Then we have that $\chi_{\liminf E_j} = \liminf \chi_{E_j}$. You should check this (the proof is short).
Thus we have by Fatou's Lemma,
$\mu(\liminf E_j) = \int \chi_{\liminf E_j} d \mu = \int \liminf \chi_{E_j} d \mu \leq \liminf \int \chi_{E_j} d \mu = \liminf \mu(E_j)$
For the second inequality, the idea is similar, but first prove and apply the reverse of Fatou's Lemma: http://en.wikipedia.org/wiki/Fatou%27s_lemma#Reverse_Fatou_lemma
