(adsbygoogle = window.adsbygoogle || []).push({}); "Except on a set of measure epsilon" vs "Almost Everywhere"

There are certain results in analysis which say that a property P holds everywhere except on a set of arbitrarily small measure. In other words, for any epsilon you can find a set F of measure less than epsilon such that P holds everywhere except possibly on F. For convenience, I'll say that such a property P holds "almost almost everywhere." For instance, a measurable function is continuous almost almost everywhere, and a convergent sequence of measurable functions is uniformly convergent almost almost everywhere.

My question is, why aren't you allowed to go from "almost almost everywhere" to "almost everywhere"? Intuitively, this is how I'd expect things to work. For each k, let F_k be a set of measure less than 1/k, such that P holds everywhere except F_k. Then since the set Z of points on which P does not hold is a subset of F_k for each k, it follows that Z is a subset of the intersection of all the F_k. But clearly this intersection has measure zero, so Z has measure zero as well. Where is the error in my reasoning?

Any help would be greatly appreciated.

Thank You in Advance.

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# Except on a set of measure epsilon vs Almost Everywhere

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