Vtyjkyuk

Suppose that we have two functions $f(x)$ and $g(x)$. We know that $f(x) ne g(x)$ on the interval, except for measure zero subsets of $[a,b]$. Assume that both functions are positive everywhere and both are strictly increasing almost everywhere (i.e. there is a subset $S$ of the interval $[a,b]$, where $[a,b] setminus S$ has measure $0$, and $f, g$ are strictly increasing on $S$).

Under these conditions on $f$ and $g$ (positive, strictly increasing a.e.), is it possible that there exists a non-constant polynomial transformation $P$ such that $P(f(x)) = P(g(x))$ almost everywhere on $[a,b]$?

I think that the following is a counterexample:

Let $f(x)=sin(x), g(x)=-cos(x)$ on $[0, fracpi2]$ and $P(X)=X^2(1-X^2)$.

Then
$$P(f(x))=P(g(x))=sin^2(x) cos^2(x)$$

It is possible to have positive, strictly increasing functions $f, g: [a,b] to mathbbR$ such that $f(x) ne g(x)$ for all $x in [a,b]$, yet there is a non-constant polynomial $P$ such that $P(f(x)) = P(g(x))$ for all $x in [a,b]$.

To find an example, let's start with the polynomial $P$: take $P(X) = (X - 2)^3 - (X - 2)$.
Then, plot the points $(X,Y)$ where $P(X) = P(Y)$. We get:

The key is to look for a point on the graph where there is a tangent line with positive slope in the first quadrant. Above, we see this occurs at the point $(3,1)$. So we can set $(f(t), g(t))$ to be a parametrization of the curve starting from this point and moving in the positive $x$- and $y$-directions. We note that by construction, $f$ and $g$ are increasing and positive, and they satisfy $P(f(t)) = P(g(t))$.

Explicitly, if we parametrize the above curve we get the following example, which also uses trigonometry like N. S.'s nice answer:
beginalign*
P(X) &= (x-2)^3 - (x-2) \
f(x) &= 2 + frac1sqrt3 cos x + sin x \
g(x) &= 2 + frac1sqrt3 cos x - sin x
endalign*
These are always positive and are both strictly increasing on some intervals, e.g. on the interval $[-2pi/3,-pi/3]$.
The algebra is a bit messy, but you can verify that $P(f(x)) = Q(f(x))$ with Wolfram Alpha.

In general, for any polynomial $P$ we can plot the set of $(X,Y)$ coordinates where $P(X) = P(Y)$, and anywhere where the tangent curve has positive slope we can read off a pair of functions $f, g$ satisfying $P(f(t)) = P(g(t))$.

Of course, there are also examples where no such polynomial exists. For instance, take
beginalign*
f(x) &= x^4 \
g(x) &= x^2
endalign*

on the interval $[1,2]$.

Note that for either example, you can make the function increasing on all of $mathbbR$ and not just $[a,b]$, by extending $f$ appropriately to values $x < a$ and $x > b$.