Question About Definition of Almost Everywhere I suppose I'm a bit confused about the definition in the following regard:
A property holds a.e. if it holds everywhere except for a set of measure $0$. Now, if the particular property is only defined for a set of measure $0$, is it a.e. by default?  
Say I have two 'continuous' (standard topology) sequences $f,g: \mathbb{N} \to \mathbb{R}.$ Are we then allowed to say that $f = g$ a.e.? Or instead do the functions have to be defined on a set of non-zero measure and a.e. refers to some measure $0$ subset?   
I ask because a homework exercise asks me if two real functions are continuous and agree a.e. on a subset of $\mathbb{R}$, are necessarily identically equal. Clearly this is true if the points are not isolated, since if continuous functions disagree at some point, they must disagree on a non-zero measure set since open sets have non-zero measure. Though it need not be if I just use sequences. 
So, what are the requirements to use the phrase a.e.?   
Wolfram definition: A property of $X$ is said to hold almost everywhere if the set of points in $X$ where this property fails is contained in a set that has measure zero. 
This would seem to imply that it is a.e. by default.

Here's an example to show that your concern about isolated points is justified.
Let $f(x)=0$ for all $x\in{\Bbb R}$ and $g(x) = \max(x,0)$. These two functions are continuous and the are equal at (Lebesgue) a.e. point of $B:=(-\infty,0]\cup\{1\}$. But they are not identically equal on $B$.
