Consider
$\int_0^1 \log(x) dx = -1$,
where the answer $1$ was obtained by using the fundamental theorem of calculus - taking the antiderivative of $\log(x)$ and then evaluating at the endpoints.
But $\log (x)$ is an unbounded function on the interval $[0,1]$ so the lower Riemann sum will be $-\infty$ and hence the integral doesn't even exist.
So does mean that whenever we are using the fundamental theorem of calculus we are making the assumption that we are dealing with Lebesgue integration instead of Riemann integration?
It should be interpreted as an improper Riemann integral: $$ \lim_{\epsilon\to 0^+} \int_\epsilon^1 \ln x \, dx . $$ For each $\epsilon \in (0,1]$, the lower Riemann sum is finite.