Is $g\circ f$ integrable if $f$ is increasing and $g$ is integrable

Question

Let $f:[a, b] \to \Bbb R$ be increasing and $g:f[a, b] \to \Bbb R$ be integrable. Is $g\circ f$ integrable on $[a, b]$?

It seems integrable to me since $g$ is integrable on $[f(a),f(b)]$ since for any partition $P'$ of $[a,b]$, we can choose partition $P''$ of $[f(a),f(b)]$ such that $$U(g\circ f,P')−L(g\circ f,P')≤U(g,P'')−L(g,P'')$$ Is this reasoning correct?

You can show without any heavy machinery that an increasing (or any monotone) function is integrable. Does that help? — Boshu, Oct 13 '18 at 08:52
@SabyasachiBasu It doesn't, $g \circ f$ can be not Riemann integrable even if $g$ is integrable and $f$ is continuous. — Adayah, Oct 13 '18 at 08:53
@Adayah yeah you're right, it's fairly easy to come up with a counter example. — Boshu, Oct 13 '18 at 08:55
@SabyasachiBasu I can show that $f\circ g$ may not be integrable. — Hash Nuke, Oct 13 '18 at 08:59
@SaucyO'Path It's given here $f\circ g$ is not integrable where $f$ is increasing and $g$ is integrable. — Hash Nuke, Oct 13 '18 at 09:02
@HashNuke Perhaps, but the bottom line of the discussion is that your problem has $g\circ f$. — , Oct 13 '18 at 09:08
@SaucyO'Path Yup, i get the feeling that this is integrable. — Hash Nuke, Oct 13 '18 at 09:09
User Boshu also got you the reading that it may not be, but feel free. — , Oct 13 '18 at 09:14

score 1 · Answer 1 · answered Oct 13 '18 at 12:05

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$$f:[0,1]\to[0,1] ,\quad f(x) = x^2 $$ $$ g: [0,1] \to \mathbb R ,\quad g = \begin{cases} \frac{1}{\sqrt x} & x\neq 0\\ 0 & x=0 \end{cases} $$

Notice that $f$ is increasing, $g$ is integrable, while $g\circ f$ is not integrable.

answered Oct 13 '18 at 12:05

Calvin Khor

34,903

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$g$ is not bounded and according my definition of my book, Riemann integrability only applied to bounded function. This is from Abbott Understanding Analysis page 227 – Hash Nuke Oct 13 '18 at 15:01
@HashNuke fair, but it seems UserS has edited your question to add measure-theory as a tag. If you don't agree with him, you should undo his edit. – Calvin Khor Oct 13 '18 at 15:02

Is $g\circ f$ integrable if $f$ is increasing and $g$ is integrable

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