So there is this function f defined by f (x) = 1 if x is rational and f (x) = 0 if x is irrational.
Is it possible to calculate the area under f(x) in the interval [0, 1]? If it exists. I know that using Riemann Integral would fail though, can we use the Legesgue Integral to do it? If so, how?
P.S. This question is originally from a Hong Kong forum LIHKG. Why would a forum post a mathematics question?
Hint: your function is equal to the function that is constantly zero almost everywhere with respect to Lebesgue measure.
Added: If $f$ and $g$ are both Lebesgue measurable, integrable functions and are equal almost everywhere with respect to Lebesgue measure then $$\int f d\mu = \int g d\mu,$$ where $\mu$ is Lebesgue measure.