Ask Mathematics

Is there a difference between those two notions of “almost everywhere”?

Assume we have two functions $f$ and $g$ on, say, the interval $[0,1]$. Let’s say that $g$ is continuous, i.e. the pointwise evaluation $g(x)$ makes sense. $f$, on the other hand, is only $L^1([0,1])$ and thus not defined pointwise.

Is there a difference between saying

a) $f = g$ a.e. on [0,1]$ and

b) There is a fixed negligible set $N$ such that $f(x) = g(x)$ for all $x\not \in N$?

On first glance it seems that b) is only the expanded definition of a) (as defined, e.g., on Wikipedia), but isn’t b) stronger than a)?

I would reason that in the setting of b), for any point outside $N$ we can now give $f(x)$ proper meaning (as being defined by $g(x)$), while in the setting a) we can never tack down any pointwise evaluation because we can always change any arbitrary point’s image $f(x)$ to any value with a) still being valid.

If there is indeed a difference, is there a name for b) (or a shorter way of writing it)?

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