Sometimes definite integral is defined using antiderivatives:

$$\int_{a}^b{f(t)dt}=F(b)-F(a)$$ where $F$ is any continuous function such that: $$(\forall t\in[a,b]\setminus C)(F'(t) \space exists \space and\space F'(t)=f(t))$$ where $C$ is a countable set. Then (if I have written the definition correctly) it can be proved, the integral is well-defined.

some not lebesgue integrable functions are integrable with this definition.

1. are all lebesgue integrable functions, integrable with above definition?

2. can this definition be extended to measure spaces?

3. will this definition be valid if we require $C$ have zero measure only (not necessarily countable)?

4. Has this integral a formal name? (eg Dieudonné integral)