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Lebesgue Measures and Integrals : Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that...

Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that:
(i) f_n -> f in [0,1] and
(ii) integral over [0,1] of f_n =< K for all n and some constant K.
Then f is in L^1[0,1] and || f||_1 =< K.
All integrals are with respect to Lebesgue measure.

Solution Preview

Proof:
We can use the Dominated Convergence Theorem(DCT).
Let g(x)=K. From the ...

Solution Summary

Lebesgue Measures and Integrals are investigated.

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