Weierstrass M Test and Lebesgue’s Dominated Convergence Theorem

Previously, we wrote a blog post about Weierstrass M Test. It turns out Weierstrass M Test is a special case of Lebesgue’s Dominated Convergence Theorem, a very powerful theorem in Measure Theory, where the measure is taken to be the counting measure.

Lebesgue Dominated Convergence Theorem: Let (f_n) be a sequence of integrable functions which converges a.e. to a real-valued measurable function f. Suppose that there exists an integrable function g such that |f_n|\leq g for all n. Then, f is integrable and \displaystyle \int f d\mu=\lim_n \int f_n d\mu.

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