First off, h(E) is measurable as h^-1 is continuous. And you need it again in the final equality and you also need the bijection. Think about the definition of the Lebesgue measure, you need the infemum, over all open intervals.
As for your argument, you are using the fact that it is a bijection to have that as for every point x you need a unique y such that (otherwise the statement may not be true).
I also suggest that you write because the I_k are sets and the sum notation does not make any sense (in the old days some people used it for union).