How do I prove this proposition from Royden's Real Analysis:

> 
> If $\mu$ is a complete measure and $f$ is a measurable function, then
>  $f=g$ almost everywhere implies $g$ is measurable.
> 
> 
> 

In proving this proposition, what differs from the proof of a proposition from the first chapters stating: 

> 
> If $f$ is a measurable function $f=g$ almost everywhere then 
>  $g$ is measurable.
> 
> 
> 

In particular, what has to be modified in the following proof:

Take $E=\lbrace x \in X | f(x) \neq g(x) \rbrace,$ which is measurable and has measure $0$. For a measurable set $A$ in the range of $g$, we show that the set $Y=g^{-1}(A)$ is measurable. Now, $Y \cap E$ has is measurable with measure $0$. Since $Y \setminus E = f^{-1}(A) \setminus E$ is a difference of two measurable sets, we are done.