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…
