1. The problem statement, all variables and given/known data Let h(x) = 0 for all x in [a,b] except for on a set of measure zero. Show that if [itex] \int_a^b h(x) \, dx[/itex] exists it equals 0. We are given the hint that a subset of a set of measure zero also has measure 0. 2. Relevant equations We've discussed the Lebesgue integrability criterion: A bounded function f is Riemann integrable if and only if f the points of discontinuity on [a,b] are a null set. 3. The attempt at a solution First, is there a case where this integral would not exist? It seems like if h is not 0 only on a null set then it would be bounded and thus integrable by the criterion. Second, I understand intuitively that the integral is 0, but I am having trouble formalizing it. If h is equal to a non-zero [itex]k[/itex] only at one point then we could set the norm of the partition, [itex]||P|| < \epsilon/k[/itex] and we would have [itex]|\sigma - 0| < \epsilon[/itex]. This could be modified for a finite number of non-zero points, by letting [itex]k[/itex] be the max of the absolute values of the non-zero points. But what if there are countably infinite non-zero points? I also haven't used the hint. Is this to be applied over partitions of [a,b]?