This problem is an exercise in unraveling the definitions of "almost everywhere" and "measurable." Let's start by listing what information the problem gives.
- $f = g$ almost everywhere means that the set $A := \{x : g(x)\neq f(x)\}$ has measure $0$.
- $f = h$ almost everywhere means that the set $B := \{x : h(x)\neq f(x)\}$ has measure $0$.
The problem then asks about the relationship between $g$ and $h$, so it is natural to consider the set $C := \{x : g(x)\neq h(x)\}$. Step 1 should be to answer the following questions: How does $C$ relate to $A$ and $B$? What is the measure of $C$?
Now suppose that $g$ is measurable. This means $g^{-1}(I)$ is measurable for every (possible infinite) interval $I$. You next want to answer the question: How does the set $h^{-1}(I)$ relate to $g^{-1}(I)$ and $C$? You want to conclude that $h^{-1}(I)$ is measurable as well.
Hope that helps some.