$f(x)$ is positive and integrable on $[a,b]$. Does that imply $\int _a^x f(t)dt$ is positive for $x\in (a,b]$?

Question

$f(x)$ is positive and integrable on $[a,b]$. Does that imply $\int _a^x f(t)dt$ is positive for $x\in (a,b]$?

Intuitively I think it's true, but don't know how to prove.

Answer 1

Yes, this is true though it is difficult to prove using methods from calculus unless you are allowed to assume that $f$ is continuous. If $f$ is continuous, then for some $\epsilon > 0$ (and wlog $\epsilon < x - a$), we have $f(t) > f(a)/2$ for all $t \in [a, a+\epsilon]$. Then $$\int^x_a f(t) dt \ge \int^{a+\epsilon}_a f(t) dt \ge \int^{a+\epsilon}_a f(a)/2 dt = \epsilon f(a)/2 > 0$$ since $f(a)$ is positive.

Otherwise, if $f$ is only measurable, you should consider the sets $A_n = \{ t \in [a,x] : f(t) > 1/n\}$. Since $f$ is positive on $[a,x]$, one of these sets must have positive measure upon which $$\int_{a}^x f(t) dt \ge \int_{A_n} f(t) dt \ge \int_{A_n} \frac 1 n dt = \lvert A_n \rvert/n > 0.$$