Positive function

[Tolaso J Kos]

Let \( f \) be a function defined on an intervel \( \mathcal {A} \) and holds: \( f(t)>0 \;\; \forall t \in (0, x ) \).

Prove that the function \( g(x)=\displaystyle \int_0^x f(t) \,dt \) is positive.

[Demetres]

If this is a Riemann integral, we use the fact that any Riemann integrable function has a point of continuity. (In fact the set of point of discontinuity has measure zero - this is Lebesgue's criterion.) So there is an interval \(I \subseteq [0,x]\) and a positive real number \(a\) such that \(f(t) \geqslant a\) for every \(t \in I\). The result follows.

For the Lebesgue integral we just consider the sets \(A_n = \{t\in[0,x]:f(t) > 1/n\}.\) We have countably many such sets with their union being equal to \([0,x].\) So \(\mu(A_n) > 0\) for some \(n\) and the result follows as the integral is at least \(\mu(A_n)/n.\)