Hello, I got a question about a lemma on Lebesgue integration (Riesz-Nagy approach).(adsbygoogle = window.adsbygoogle || []).push({});

Let f(x) be a Lebesgue integrable function on interval (a, b). Riesz and Nagy (pg. 50 of Lessons of Functional Analysis) say that if f(x) is not bounded, for all epsilon > 0 we can decompose f(x) into the sum of a BOUNDED INTEGRABLE FUNCTION g(x) and a function h(x) such that

(Integral from "a" to "b") |h(x)|dx < e.

From Apostol I knew this lema (10.19.b of Mathematical Analysis): if f(x) is the limit function of a increasing sequence of step functions (therefore a Lebesgue integrable function), for all epsilon > 0 we can decompose f(x) into the sum of a STEP FUNCTION s(x) and a function h(x) such that

(Integral from "a" to "b") |h(x)|dx < e.

I would thank if anyone can give me hints to pass from Apostol lemma to Riesz lemma.

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# Lemma of Lebesgue integration

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