Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 25J / a / Solution

An $\mathcal{A}$-simple function is a measurable function taking finitely many values, written $s={\sum }_{j=1}^m{a}_j{1}_{A_j}$ with disjoint measurable $A_j$. For nonnegative simple functions define $\int sd\mu ={\sum }_j{a}_j\mu (A_j)$, with $0\cdot \infty =0$. Refining two disjoint descriptions shows this value is independent of the description. For nonnegative measurable $h$, defineFor a real measurable $f$, put $f^{+}=\max (f,0)$ and $f^{-}=\max (-f,0)$. It is Lebesgue integrable when $\int |f|d\mu <\infty$, equivalently both positive and negative parts have finite integrals. ThenThis includes Lebesgue integrable signed simple functions, whose integral is the same finite weighted sum.