Hi everyone, I'm struggling to prove the following two results:

Let f, g : R -> R be continuous real-valued functions on R such that f = g

lambda-almost everywhere (f = g everywhere except on a set of Lebesgue measure 0). Then f = g everywhere.

Let f : [a, b] -> R be a strictly increasing function on [a, b]. Then f is

Lebesgue measurable.

Can someone give me a hint on where to go with this?

Thanks.