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Let f be a non-negative Lebesgue measurable function with $\int_\mu\ f \,d\lambda< \infty$. Show that for each $\epsilon>0$, there exists a $\delta>0$ such that $\lambda(E)<\delta$ implies $\int_E\ f \,d\lambda< \epsilon$.
--I think I need to use the simple function $\phi=f\chi_E$ such that $\phi<f$ but I do not see how to tie that into the standard epsilon-delta proof with a finite l Lebesgue integral.
--I also know that $\lambda(A_c)\leqslant(1/c)\int_E\ f \,d\lambda$ from the previous problem so I am assuming I will need to use that.