Proof that the Integral of a Positive Function is Positive?

Solution 1:

This doesn't make sense for the Riemann integral, since a function which is any worse than almost-everywhere continuous (that is, the set of discontinuities has positive measure) fails to be Riemann integrable.

For the Lebesgue integral, any measurable function $f : \Bbb R \to \Bbb R$ which is positive on a set of positive measure (i.e. a nonnegative function which is not identified with the zero function) has positive Lebesgue integral. To see this, note that at least one (actually, all but finitely many) of the sets $\{x : f(x) > \frac 1 n\}$ has positive measure; hence $$\int_{\mathbb R} f \ge \int_{\{x : f(x) > \frac 1 n\}} f \ge \frac 1 n m\left\{x : f(x) > \frac 1 n\right\} > 0$$ where $m$ is Lebesgue measure.

So by the two "main" (or at least, arguably the most common) definitions of the integral, any function which is integrable, not negative, and not always zero, has positive integral.