How do you show that a monotone increasing real function on a closed interval [a, b]

Hello Everyone,

I have two questions on measure theory:

1)

How do you show that a monotone increasing real function on a closed interval [a, b] is Lebesgue measurable function.

2)

Suppose A is a subset of the closed interval [0,1] which is dense. (That is, the closure of A is all of [0,1].) suppose U is an open set with A⊆U. Must it be true that m(U) ≥ 1?

Thank you!

Is integrability of a function means that the function Lebesgue measurable ?

If a function is integrable, does it mean that the function **Lebesgue** measurable ?

Thank you very much.