(1) Define what it means for a set $Asubset bR^2$ to have zero content.
(2) Prove the following result: Let $g:[a,b] obR$ be bounded and integrable. Show that its graph $$ex graph(g)=sed{(x,g(x));xin[a,b]} eex$$ has zero content.
Proof:
(1) If $$ex infsed{sum_{i=1}^infty |I_i|; Asubset cup_{i=1}^infty I_i}=0, eex$$ then $A$ is said to have zero content. Here, $sed{I_i}_{i=1}^infty$ are rectangles with $|I_i|$ being their areas.
(2) Since $g$ is (Riemann) integrable, we have $$ex lim_{sen{T} o 0}sum_{i=1}^n (M_i-m_i)lap x_i=0, eex$$ where $$ex T: a=x_0<x_1<cdots<x_n=b, eex$$ $$ex sen{T}=max_i lap x_i, lap x_i=x_i-x_{i-1}, eex$$ $$ex M_i=sup_{xin [x_{i-1},x_i]}f(x),quad m_i=inf_{xin [x_{i-1},x_i]}f(x). eex$$ Thus (by the $ve-del$ definition of limit), $$ex forall ve>0, exists T,st graph(f)subset cup_{i=1}^n [x_{i-1},x_i] imes [m_i,M_i], eex$$ $$ex |[x_{i-1},x_i] imes [m_i,M_i]| =sum_{i=1}^n (M_i-m_i)lap x_i<ve. eex$$ Consequently, $$ex infsed{sum_{i=1}^infty |I_i|; graph(f)subset cup_{i=1}^infty I_i}=0 eex$$ This yields the desired result.