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I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is quite non-standard.

Namely, a function $f$ is said to be integrable if there exists a sequence of integrable simple functions functions $\{f_n\}$ such that a) $f_n\to f$ a.e. and b) $\{f_n\}$ is Cauchy in the mean. (It is also shown that we may replace a) with a') $\{f_n\} \to f$ in measure.) And the integral is defined to be $$\int f d\mu = \lim_{n\to \infty} \int f_n d\mu.$$

Note,the integral of a simple function, $s=\sum_1^n\alpha_i \chi_{E_i}$ is here defined to be $\sum_1^n \alpha_i \mu(E_i)$ (without taking any suprema).

I've noticed on occasion that I much prefer the standard definition where the supremum is taken over the set of integrals of all simple functions $0\leq s \leq f$ (cf. grown up Rudin), which makes me wonder: How do they relate?

I'm not so much interested in a proof of equivalence but rather some hand-waving explanation why these should be the same (if so) and an explanation as to why one would prefer to define it one way or the other - what are the advantages? I.e. I'm trying to gather some gut-feeling.

Kind Regards and thanks in advance!

ZMI

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    $\begingroup$ I don't suppose you meant to talk about begging the question. You also seem to have forgotten to include the definition in question: I guess that would be $\lim\int f_n$? $\endgroup$ – tomasz Oct 7 '14 at 11:02
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    $\begingroup$ Terribly sorry. I will that fix instantly. Oh, didn't know that was a "loaded concept" I'll correct for clarity. $\endgroup$ – Winston Oct 7 '14 at 11:24

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