Here is a problems after the measure theory section.
Suppose {$\alpha_j$} $\subset (0,1)$.
a. $\prod $(1-$\alpha_j$) > 0 iff $\sum \alpha_j < \infty $. (Compare $\sum log(1- \alpha_j) to \sum \alpha_j$.)
b. Given $\beta \in (0,1)$, exhibit a sequence ${\alpha_j}$ such that $\prod(1-\alpha_j) = \beta$.
The problem is interesting. The infinite product gives the measure of the generalized Cantor set.
Any idea about how to prove it?
Hint: For the only if direction in part (a), you can pretty easily check that $\log(1-a_j) < -a_j$ when $0<a_j<1$ which, lucky for you, is always the case. Once you have that, do the comparison that is suggested in the problem statement. You'll need to "undo" the logarithm at some point, too.