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Continuity and Outer Measure

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=<y=< f(x)}

Prove that; m*(A) = integral from 0 to 1/2 of f(x)dx.

Please I don't want a solution from a book, I want a simple proof based on basic definitionsand given info.

Solution Summary

Continuity and outer measurability are investigated. The solution is detailed and well presented.