Measure theory qualifying exam questions

Question

The following question is a qualifying exam question, though I don't see how these two parts are related.

(a)Let $(X, \mathcal F, \mu)$ be a measure space with $\mu(X)=1$ and suppose $F_1 , F_2, ...F_7$ are 7 measurable sets with $\mu(F_j) \geq 1/2$. Show that there exist indices $i_1<i_2<i_3<i_4$ for which $F_{i_{1}} \cap F_{i_{2}} \cap F_{i_{3}} \cap F_{i_{4}} \neq \varnothing $.

Thoughts: It seems this one is obvious I just suppose it's empty set then I can get a contradiction since the measure of the whole set is 1. I'm not sure whether it is the right track.

(b) Let $m$ denote Lebesgue measure on $[0,1]$ and let $f_n \in L^1 (m)$ be nonnegative and measurable with $\int_{[0,1/n]} f_n dm \geq 1/2$ for all $n \geq 1$. Show that $\int_{[0,1/n]} [\sup_{n} f_n] m(dx)=\infty$.

Thoughts: For this one, first I assume the integral of the sup is a finite number $M$ which is not infinity, then I can get the integral of the sup is going to zero,rather than $\geq 1/2$. I'm not sure about this idea either.

Any comments would be appreciated.

Edit: Thanks you guys. For (a), I'm wondering whether three sets also satifies the same result. I don't see why there always 4 such sets.

Answer 1

For part (a): $$\int \sum_{j=1}^7\chi_{F_j}\ge 7/2>3.$$Since $\mu(X)=1$ there must exist $x$ with $\sum\chi_{F_j}(x)>3$.

Answer 2

For Part (b), you can also do the following: Suppose that $g =\sup f_n$ is integrable. By the dominated convergence theorem, this yields $$ \int_{[0,1/n]} g \, dx \to 0, $$ but we have $$ \int_{[0,1/n]} g \, dx \geq \int_{[0,1/n]} f_n \, dx \geq 1/2, $$ a contradiction.