I am currently looking through Bartle's The Elements of Integration and Lebesgue Measure, and am reading the proof of the following theorem:
If $F_{k}$ is a decreasing sequence of Lebesgue measurable sets, and if $m(F_{1}) < \infty$, then $\displaystyle \left(\cap_{k=1}^{\infty}F_{k}\right) = \lim_{k \to \infty} m(F_{k})$.
The proof defines $E_{k}:=F_{1}-F_{k}$ for $k \in \mathbb{N}$, "so that $\{E_{k}\}$ is an increasing sequence of measurable sets". I find this a bit confusing: Essentially, this sequence consists of everything that is in $F_{1}$ but not in the subsequent $F_{k}$'s. Could somebody please explain to me why this is an increasing sequence?
Also, why is the $m(F_{1}) < \infty$ necessary? Is that just to make sure it's increasing (not sure whether Bartle uses "increasing" to mean "strictly increasing" or just "nondecreasing")?
Since $\{F_k\}$ is a decreasing sequence, $F_k\supset F_{k+1}$. So $F_{k}^c\subset F_{k+1}^c$ and $$ E_k=F_1-F_k=F_1\cap F_k^c\subset F_1\cap F_{k+1}^c=F_1-F_{k+1}=E_{k+1} $$ So $\{E_k\}$ is a increasing sequence.