Let $(X,\mathcal{A},\mu)$ be a measure space and $u$ some measurable function. If $v$ is a function and $u=v$ a.e.; when is true to say that $v$ is measurable?
Also let P be some property, and $f$ & $g$ measurable functions and that property P holds for $f$ a.e.. If $f=g$ a.e.; when is true (if ever) to say that P holds for $g$ a.e.?
Any feedback on these two questions are greatly appreciated. Thanks
Equality $u=v$ a.e. means that $v$ is equal to $u$ outside a set of measure zero, a null set. For $v$ to be measurable you need not just this null set to be measurable, but also any of its subsets. They may not be, and then $v$ may not be measurable. For example, take $u=0$ everywhere, and $v=0$ everywhere except on a non-measurable subset of a null set, where $v=1$. Then $u=v$ outside a null set, but $v$ is non-measurable.
However, the original $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$ can always be completed by adjoining subsets of null sets and assigning measure $0$ to them. Function $v$ becomes measurable in the completion.
Your second statement is correct assuming $f$ and $g$ are measurable. That $f$ has property $P$ a.e. means it has it outside of a null set, and $f=g$ a.e. means $g$ is equal to $f$ outside of another null set, so $g$ has $P$ outside the union of those two null sets, which is also a null set.
However, one has to be careful with the meaning of "property" here, above "$f$ has $P$ a.e." means "$f(x)$ has $P$ for $x$ outside a null set". Being positive is an example of $P$ that works. Being continuous or smooth, on the other hand, is not such a property. The key difference is that smoothness and continuity at $x$ can be altered without changing the value at $x$, they depend on behavior of $f$ at multiple points around it.
Lebesgue spaces like $L^1$ are designed to deal with such properties. Technically, their elements are not functions but equivalence classes of functions up to equality almost everywhere. When someone says that an $L^1$ function is continuous a.e. for example, they mean that the class of functions equal to $f$ a.e. has a continuous a.e. representative.