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