In Royden's 'Real Analysis', he first defines the Lebesgue integral of simple functions that vanish outside a set of finite measure and then extends this to non-negative bounded measurable functions that vanish outside a set of finite measure. But in Rudin's 'Real and Complex Analysis', he straightaway defines the integral of a non-negative function as the limit of integrals of simple functions. He does not include any finite measure support considerations. My question is, what is the difference between these two approaches?
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asked Nov 27, 2019 at 10:52
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1$\begingroup$ It is not necessary to consider the special case first. You c an define integrals for all non-negative simple functions and extend it to all non-negative measurable functions without worrying about finiteness of the integrals. Rudin's approach is more standard. $\endgroup$– Kavi Rama MurthyCommented Nov 27, 2019 at 12:04
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I wanted to put it as a comment, but I didn't have enough place.
A characteristic function $1_A$ is integrable $\iff$ $m(A)<\infty $.
So, probably in Rudin, they suppose that a simple function is defined as $$\sum_{i=1}^n a_i 1_{A_i}$$ where $m(A_i)<\infty $, whereas in Royden, they define a simple function as $$\sum_{i=1}^n a_i1_{A_i}$$ without assumption that $A_i$ have finite measure.
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1$\begingroup$ Wrong. Rudin does not require $m(A_i) <\infty$. See my comment above. $\endgroup$ Commented Nov 27, 2019 at 12:04
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$\begingroup$ @KaboMurphy: No problem, I will erase my comment then. I don't know this book, so it was just a (possible) explanation $\endgroup$– HermanCommented Nov 27, 2019 at 12:43