Well, the answer is pretty easy. Consider the function on arbitrary interval [a,b]. Then that function has some nonzero values only on the set of measure zero (since all rational numbers on the [a,b] is countable set, and so it's Lebesque measure is zero). But Lebesque integrals of functions having different values on the set of measure zero are equal, so you may write:
\int_{[a,b]} f(x) =\int_{[a,b]} 0 = 0.
And that's it. |
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