If $f(x)<g(x)$, can $\int_a^b f(x)\,dx = \int_a^b g(x)\,dx$?

If one function is strictly less than another at every point in a set whose measure is positive, then their integrals over that set are not equal.

Consider the set of points $x$ at which $\dfrac 1 {n+1} < g(x)-f(x) \le \dfrac 1 n$ for $n=1,2,3,\ldots$ and the set of points $x$ at which $g(x)-f(x)>1.$ The union of those sets is the whole domain, so the measure of at least one of them is more than $0.$ And the integral of $g-f$ over that set is more than $1/(n+1)$ times the measure of that subset of the domain, so it's positive.

Using Riemann's approach, this seems more complicated, although it would be easier if one had an assumption of continuity.