Maybe a dumb question based on all the questions I've asked for the last decade, but what's the general way to do the Lebesgue integral of some non-negative (measurable?) function that is Riemann integrable like, say, a non-horizontal line like $y=x, x \in [0,1]$ and not using any theorem that Lebesgue integral = Riemann integral but really by definition like below?
$$\int_E f d\mu = \sup \{\int_E s d \mu | 0 \le s \le f, \ \text{s simple}\}$$
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(I see a related post, but it seems to talk about really Riemann integrability Show Lebesgue integration by Lebesgue integral definition. )
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Specific example: Also asked here but not really any answer Computing lebesgue integral of a piecewise function – it's actually in Royden Fitzpatrick to ask about integrating $\int_0^1 f$ with $f(x)=x^{\alpha}1_{(0,1]} + 0 1_{\{0\}}$ and $\alpha \in \mathbb R$. Ok so for $\alpha =1$, how to do this by definition?
Guess: I think of doing step functions like $s_2 = \frac121_{[\frac12,1]}$, $s_3 = \frac131_{[\frac13,\frac23]} + \frac231_{[\frac23,1]}$, and so on with $$\int s_n = \frac1n [\frac1n + \frac2n + … + \frac{n-1}n]=\frac{n-1}{2n}, n \ge 2$$ and then because $\frac{n-1}{2n}$ is increasing and bounded above by $\frac12$ monotone convergence theorem says $\sup = \lim$ and then $\lim =$, unsurprisingly, $\frac12$.
Questions:
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Is that correct? I mean…how would I know the supremum over every $s$ is the same as the supremum for these particular $s_n, n \ge 2$?
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Please provide some textbook reference that shows elementary Lebesgue integral computations like this (or where in Royden Fitzpatrick this is besides the exercises since I can't find a solutions manual).
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And now going back to the general question: How would you know Gasai what $s$ to pick, and how would you guarantee these $s$ picked somehow do cover the cases of all $s$ ? Like is there some theorem that says you can do this integral for an Ansatz simple functions?
Wait…is that what simple approximation lemma and rth staircase function are about?
Simple approximation lemma:
David Williams' Probability with Martingales rth staircase function:
Best Answer
Oh wait here
https://en.wikipedia.org/wiki/Lebesgue_integration#Non-negative_functions
I think it's the same as the rth staircase function in Probability with Martingales. $k=1$ to $n$ and $i=1$ to $r$.
I think the Royden Fitzpatrick Real Analysis simple functions are generalisations ...with their $k$ = PWM's $i$ and their $\epsilon$ = PWM's $\frac{1}{2^r}$ sooo there are several $\epsilon$'s here...just plug in each $r$...or something.
I guess my $s_n$'s are like $\epsilon = \frac1n$ but $\epsilon = \frac1{2^n}$ would work too.
And finally I'll just take for granted that $$\sup_{\text{all simple s} } \int s= \lim_{n} \int \{\text{those} \ s_n \ \text{'s}\}$$
I have a feeling for all $s$, there exists an $n$ with $0 \le s \le s_n \le f$ or something. then $\sup \int s = \sup \int s_n = \lim \int s_n$.