Let {fn} be a sequence of funtions in L[a, b]. Suppose that there exists a function g ∈ L[a, b] with |fn(x)| ≤ g(x) a.e. for each n. Show that if limn→∞ fn(x) = f(x) a.e. and h(x) is a bounded measurable function, then limn→∞ b a fnh = b a fh

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A: f is defined on the set of real numbers. The detailed solution is as follows below:

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