The definition of continuity is that takes open sets to open sets. f is a measurable function if takes open sets to measurable sets. But, every open set is measurable.

Thus, all that remains is to show that the a.e. condition doesn't effect measurability.

Define g:[a,b]-->R to be:

g(a) = f(a) if f is continuous at a

g(a) = if f is not continuous at a

Then g is a continuous function, so g is measurable. But g = f a.e. thus, f is also measurable.

Note: You may have to show g is well-defined