Suppose f is integrable and g is bounded on [a,b], and g differs from f only at points in a set H with the following property: For each $\epsilon>0$, H can be covered by a finite number of closed subintervals of [a,b], the sum of whose lengths is less than $\epsilon$. Show that g is integrable on [a,b] and that $\int_a^b$$g(x)dx=$$\int_a^b$$f(x)dx$.
I'm not sure if I'm on the right track, but what I'm thinking is I need to show that since g is bounded on [a,b] there is a partition P of [a,b] which has intervals smaller than $\epsilon$ and that S(P)-s(P)<$\epsilon$ for g. I believe that should show that g is integrable on [a,b], but I don't know how to show $\int_a^b$$g(x)dx=$$\int_a^b$$f(x)dx$.
Thanks!