Suppose I have an integral:
$F \equiv \int_{a}^{b} f(x) dx$
I know that $f(x)$ is real-valued, finite and non-negative everywhere on the interval $(a, b)$ where $a$, $b$ are real numbers or $\pm \infty$ and $a \leq b$. I know that in the obvious, well-behaved cases this implies $F \geq 0$.
Is there any obscure pathological case where $F < 0$? You may generalize to cases that involve double, triple or higher order integrals. However, for each integral the property that the upper bound is greater than or equal to the lower bound and the bounds are always real numbers or $\pm \infty$ must hold.