c) Use Lebesgue's dominated convergence theorem to prove the following result: Let L(I) be the set...

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c) Use Lebesgue's dominated convergence theorem to prove the following result: Let L(I) be the set of Lebesgue functions on an interval I and {g} be a sequence of functions such that the following holds. i) each g is non-negative almost everywhere on I, ii) the series [g, converges almost everywhere on I to a function g which is n=1 bounded above by a function in L(I). Then g = L(I), the series Σfg, converges, and we have ſΣgn =] gn 'n n=1 1 n=1 n=1 I c) Use Lebesgue's dominated convergence theorem to prove the following result: Let L(I) be the set of Lebesgue functions on an interval I and {g} be a sequence of functions such that the following holds. i) each g is non-negative almost everywhere on I, ii) the series [g, converges almost everywhere on I to a function g which is n=1 bounded above by a function in L(I). Then g = L(I), the series Σfg, converges, and we have ſΣgn =] gn 'n n=1 1 n=1 n=1 I