The number of summands for each simple function may be different. So the $k$ may depend on your $n$.

I think it is better to try doing something like this. First, it is sufficient to just assume that and . Let E be the set were f is positive, we want to show that E is a set of measure zero. One way of doing this is to use one of the limit theorems on measures. Define to be the set were and argue that each has measure zero, from here it will follow that E has measure zero.