Problem 1472 (difficulty: 2/10)

Let \(\displaystyle A\subset\R\) be Lebesgue-measurable and \(\displaystyle \chi_A(x)=\begin{cases}1&x\in A\\ 0 & x\not\in A\end{cases}\). Show that \(\displaystyle \int_{\R}\chi_A \,\mathrm{d}\lambda=\lambda(A)\).

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