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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)?