Use the concept of Baire Category to prove that the upper integral of a positive function is always positive.

Question

Use Baire Category to prove that the upper integral from $0$ to $1$ of any $f:[0,1]\to(0,1]$ is greater than $0$.

Answer 1

Hint: Given $n\ge 1$, let $E_n$ be the closure of the set $\{x\in[0,1],f(x)\ge\frac{1}{n}\}$. Apply Baire's category theorem to $\cup_{n=1}^\infty E_n=[0,1]$.