# Composition of two absolute functions

$f$ and $g$ are two absolutely contiunous functions and $f$ is monotone.

Is $f(g)$ necessarily an absolutely continuous function? Why not any counter example?

Thanks a lot!!

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What did you try? – Davide Giraudo Sep 13 '12 at 8:54
Refer to the book Measure Theory by V.I. Bogachev. There is hint for your problem. Try online books. :) – Babak S. Sep 13 '12 at 9:25
Babak I am curious about f(g) not g(f) thanks for the book though.. Davide I get stuck in the second stage when composed because disjointness criterion gets lost... – Salih Ucan Sep 14 '12 at 1:29

Let $\{x_n\}_1^\infty$, $\{y_n\}_1^\infty$ be two sequences of point in $I = [0, 1]$ such that $$x_1 = 0\\ x_n\to 1\\ x_n < y_n < x_{n+1} \quad n=1, 2, \dotsc$$ Let's define a function $h$ on the interval I setting $$h(x) = \begin{cases} \frac 1 {(y_n - x_n)(n + 1)^2} &x\in [x_n, y_n)\\ -\frac 1 {(x_{n + 1} - y_n)(n + 1)^2} &x\in [y_n, x_{n + 1}) \end{cases}$$ $h$ is summable \begin{align} \int_I \lvert h(x)\rvert dx &= \sum_{n = 1}^\infty \frac {y_n - x_n} {(y_n - x_n)(n + 1)^2} + \sum_{n = 1}^\infty \frac {x_{n + 1} - y_n} {(x_{n + 1} - y_n)(n + 1)^2} \\ &= 2\sum_{n = 1}^\infty \frac 1 {(n + 1)^2} \\ &= \pi^2/3 - 2 \end{align} Of course $$f(x) := \int_0^x h(t) dt, \quad x\in I$$ is absolutely continuous, moreover $$f(x_k) = \int_0^{x_k} h(t) dt = \sum_{n = 1}^{k - 1} \frac 1 {(n + 1)^2} - \sum_{n = 1}^{k - 1} \frac 1 {(n + 1)^2} = 0\\ f(y_k) = \int_0^{y_k} h(t) dt = \sum_{n = 1}^{k} \frac 1 {(n + 1)^2} - \sum_{n = 1}^{k - 1} \frac 1 {(n + 1)^2} = \frac 1 {(n + 1)^2}$$ From the above equalities we deduce the the total variation $V$ of the function $\sqrt f$ isn't finite $$V \geq \sum_{n = 1}^{\infty} \left\vert \sqrt {f(y_n)} - \sqrt{f(x_n)} \right\vert = \sum_{n = 1}^{\infty} \frac 1 {n + 1} = \infty$$ In particular $\sqrt f$ is not absolutely continuous even if it is the composition of two absolutely continuous functions, $f$ and $$g: x\in I \to \sqrt x \in I$$
Same idea as @AlbertH, but a bit simpler to write: $f(x)=x^2 \sin^2 \frac{\pi }{2x}$ is Lipschitz on $[0,1]$ because $f'$ is bounded. Hence, $f$ is absolutely continuous. However, $\sqrt{f}$ has infinite variation on $[0,1]$, which one can demonstrate by summing $|f(\frac1n)-f(\frac1{n+1})|$.