1.
Let $ f,g$ be real-valued integrable functions on a measure space $ (X,B,\mu)$,and define:
$ F_t=\{x\in X:f(x>t)\},G_t=\{x\in X:g(x)>t\}$.
Prove:
$ \int|f-g|d\mu=\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$.
proof:
by Fubini theorem(cake representation theorem in fact):
$ \int|f-g|=\int_{0}^{\infty}\mu(\{x||f-g|(x)>t\})dt\\
=\int_{-\infty}^{\infty}\mu(\{x|f(x)>t>g(x)\})+\mu(\{x|f(x)<t<g(x)\})dt\\
=\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$.
(there is a geometric heuristic,strict proof is due to fubini theorem)
Q.E.D.
2.
Let $ H$ be a infinite dimensional real Hilbert space.
a)Prove the unit sphere $ \{x\in H:||x||=1\}$ of $ H$ is weakly dense in the unit ball $ B=\{x\in H :||x||\leq 1\}$ of $ H$.
b)Prove there is a sequence $ T_n$ of bounded linear operator from $ H$ to $ H$ such that $ ||T_n||=1$ for all n but $ lim T_n(x)=0$ for all $ x\in H$.
proof:
by Zorn lemma there is a orthogonal bases $ \{e_i\}$.
to proof a),suffice to proof:$ \forall x,\exists x_n,\forall y\in H,\lim_{n \to \infty}<x_n,y>=<x,y>$.
this can be done by look at the expansion $ y=\sum_{i}<y,e_i>e_i$.due to the Cauchy inequality,there is a freedom of choice the coefficient $ <e_i,x_n>$ for $ i>>n$.the choice will lead a).
b) is trivial due to a).
Q.E.D.