Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions

Let (X,M,μ) be a measure space and suppose {fn} are non-negative measurable functions decreasing pointwise to f. Suppose also that f1<. Then Xf dμ=limnXfn dμ.


Since {fn} are decreasing, and converges pointwise to f, then {fn} is increasing pointwise to f. So by the monotone convergence theorem
Xf dμ=limnXfn dμ and so Xf dμ=limnXfn dμ.


The problem is that fn increases to f which is not non-negative, so we can’t apply directly to fn the monotone convergence theorem. But if we take gn:=f1fn, then {gn} is an increasing sequence of non-negative measurable functions, which converges pointwise to f1f. Monotone convergence theorem yields:
so limn+Xfndμ=Xfdμ.

Note that the fact that there is an integrable function in the sequence is primordial, indeed, if you take X the real line, M its Borel σ-algebra and μ the Lebesgue measure, and fn(x)={1 if xn 0 otherwise
the sequence fn decreases to 0 but Rfndλ=+ for all n.

Source : Link , Question Author : Kuku , Answer Author : T. Eskin

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