I am reading Patrick Billingsley for measure theoretic probability.I encountered the following theorem(Th $32.2$,page-$404$,Billingsley($1995$)).which states the following:
Statement: A non-decreasing function $F:[a,b]\to \mathbb R$ is differentiable almost everwhere on $(a,b)$.If $f$ is defined to be $f(x)=\begin{cases} \frac{d}{dx}F(x) , \text{if $F$ is differentiable at $x$}\\ 0 \text{, otherwise}\end{cases}$.Then $f$ is Borel-measurable,non-negative and satisfies $\int_{a}^b f(x)dx\leq F(b)-F(a)$.
It looks to me some kind of analogue to the Fundamental theorem of calculus.But I always wonder that why shouldn't we expect equality always(It seems kind of counterintuitive to me).In particular I am looking for some intuition that would convince me that this is the result one should get,not more than that.Also I would like to see an example where strict inequality holds.I think that would help me understand the theorem.
Detailed explanations from the members of MSE community will be highly appreciated.
Peek-a-boo mentioned in the comments that a piecewise constant function is a counter-example. Let me point out that equality in the Fundamental Theorem still fails even when we assume $F$ is continuous.
The standard counter-example is the so-called Devil's Staircase
This is a function that is defined to be $\frac{1}{2}$ on the first interval removed in the construction of the Cantor set, $\frac{1}{4}$ and $\frac{3}{4}$ on the next two intervals, and so-forth for all the intervals in the complement of the Cantor set. This has a continuous extension to the Cantor set itself, giving an example of a continuous monotone function whose derivative is equal to $0$ almost everywhere.
The additional condition you need to make the fundamental theorem work is known as absolute continuity. The formal definition is at the link, but the idea is you must be able to control the total variation in the output with the total variation in the input. This is necessary in order to prevent bad behavior on sets of measure $0$, which is essentially what is happening with the Devil's staircase example - all the variation happens on the Cantor set, which has $0$ measure, and so the integral can't detect it.