Q:

Positive function with zero lower integral

Give an example of a positive function $f(x):[0,1]\to(0,1]$ the lower integral of which is equal to zero.

A:

Hint: 
$$ f(x) = \begin{cases} 1 & \text{if}~ x~\text{is irrational}\\
 1/n & \text{if}~x=m/n,~\text{where}~m,n\in \mathbb N,n\gt 0,\gcd(m,n) =1
\end{cases} $$ 

