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$f(x)$ 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.

Hongyan
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  • Do you know that if $f$ is Riemann integrable, then $f$ is continuous a.e.? – zhw. Mar 04 '18 at 06:54
  • You need to show that $f$ is continuous somewhere on $[a, b] $ and therefore maintains a constant sign on some sub-interval of $[a, b] $ and therefore the integral is positive. See https://math.stackexchange.com/a/519921/72031 – Paramanand Singh Mar 04 '18 at 09:30

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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 $$\int_{a}^x f(t) dt \ge \int_{A_n} f(t) dt \ge \int_{A_n} \frac 1 n dt = \lvert A_n \rvert/n > 0.$$

User8128
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  • Thanks! I know it's true for $f$ continuous. have not learned measure theory........... – Hongyan Mar 04 '18 at 06:47
  • As a workaround: if $f$ is Riemann integrable on $[a,b]$ then $f$ has points of continuity in $[a,x]$ for any $x > a$ so you can revert back to the proof for the continuous case. – User8128 Mar 04 '18 at 06:54
  • @User8128 that's smart! – Hongyan Mar 04 '18 at 07:13
  • @Holo I don't understand the question. If $f$ is not defined on all of $A$, then there is no way to define $\int_A f , d\mu$. There is no way to "imply that $f(x)$ exists" using the value $\int_a^b f(x) dx$. Indeed, $f(x)$ must be defined at all $x \in (a,b)$ [or at least at almost all $x \in (a,b)$ if we are speaking in the measure theoretic sense] in order to even define $\int_a^b f(x) dx$. – User8128 Mar 04 '18 at 07:18