I am trying to find a Borel measurable function $f : [0, 1] \rightarrow (0, \infty)$ such that $L(f,[0,1]) = 0$ (lower Riemann sum).
I have found for example $f=\chi_{[0,1] \backslash \mathbf{Q}}$.
This $f$ has $L(f,[0,1]) = 0$ and if $\lambda$ is Lebesgue measure on the Borel subsets of $[0, 1]$, then $\int g d \lambda=1$. But, i am struggling to find an example with $f : [0, 1] \rightarrow (0, \infty)$. Any help would be greatly appreciated. Thanks
How about something like this:
If $x$ is rational and $x=a/b$ in lowest terms, let $f(x) = 1/b$. If $x$ is irrational, let $f(x) = 34$. This is supposed to have the properties: $f(x) > 0$ for all $x$, and $\inf\{f(x), a < x < b\} = 0$ for all nonempty open intervals $(a,b)$.