How to calculate a Lebesgue Integral of a function whose domain is irrational number?

Question

Let f be a Lebesgue measurable function on [0,1] and $$ f(x)=\left\{\begin{array}{ll} x^{4} & \text { if } x \in[0,1] \backslash \mathbb{Q} \\ 0 & \text { if } x \in[0,1] \cap \mathbb{Q} \end{array}\right.$$

How to calculate $\int_{[0,1]} f d m$?

Answer 1

HINT:

$f=g$ a.e., where $g(x)=x^4$ for all $x\in [0,1]$.

Answer 2

Hint:

If $f$ and $g$ are measurable functions with: $$m(\{x\mid f(x)\neq g(x)\})=0$$ and $\int fdm$ is defined then also $\int g dm$ is defined, and this with:$$\int fdm=\int gdm$$