Q on Lebesgue integrals.

In a previous problem I posted here:

Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2.
Let A = { (x,y) : 0 =< x = 1/2, 0=Prove that;
m*(A) = integral from 0 to 1/2 of f(x)dx.

Now knowing that the above is true, I want to show that the integral is lebesgue measurable, that is, the area under the curve is Lebesgue measurable.

In the previous problem it was proven that the inner and outer measures are equal to the integral, but then they are both equal, I wonder if I have to prove any more results? If so, please show me.