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Most textbooks on measure and integration theory, functional analysis and probability theory introduce Lebesgue spaces $\mathscr{L}^p\langle\varOmega,\mathcal{A},\mu\rangle$ as spaces on functions, and quickly go on to identify a.e. equal functions to form the quotient spaces $L^p\langle\varOmega,\mathcal{A},\mu\rangle$ of equivalence classes, and insist on working with those. This makes the $L^p$-norm a real norm instead of a pseudo-norm (at least for $p\geq1$), and the associated $L^p$-distance a real metric. But, what is the point of this? The only advantage that I can think of of passing to equivalence classes is that it makes $L^p$-limits unique (and making the spaces Hausdorff). But then, no one's really interested in the limiting equivalence class, and when we work with limits, we automatically pick one of the a.e. equal limiting functions. So, is the convention just for cleaner proofs, or are there important results/proofs in functional analysis that don't go through without the Hausdorff structure (which is sacrificed by a (strict) pseudo-metric)?

As far as I can tell, at least in probability theory, identifying all functions that are equal almost everywhere isn't always an option. I can't remember where exactly I read this (probably one of David Williams' books) but demands such as continuous paths for stochastic processes require us to not "completely ignore" what happens on null sets.

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  • $\begingroup$ I think it's just conventional. As you say, the only thing that changes is nonuniqueness of limits, really. You can think of a pseudometric space as being like a category (in fact they are a kind of enriched category), where points being at distance zero corresponds to objects being isomorphic. So it is the usual deal with isomorphism classes of objects vs. objects. $\endgroup$ – Qiaochu Yuan May 24 '17 at 0:13
  • $\begingroup$ What about the fundamental theorem of the calculus of variations? It's conclusion would read a bit strange without identifying a.e. $\endgroup$ – Dirk May 24 '17 at 1:45
  • $\begingroup$ @Dirk, interesting. Do you mind sharing a link to the precise formulation of this "fundamental theorem", so that I can check for myself? I studied Calculus of Variations from Gel'fand and Fomin, and do not remember any theorem labeled "fundamental". $\endgroup$ – sami.spricht.sprache May 24 '17 at 14:25
  • $\begingroup$ It's this one en.m.wikipedia.org/wiki/… If do not assume continuity a priori you'll only get that the respective function is zero a. e. $\endgroup$ – Dirk May 24 '17 at 16:02
  • $\begingroup$ @Dirk, I see. Thing is that this isn't all that different from what Qiaochu Yuan says above --- convention. The theorem will, in your words, "read a bit strange" but its functionality will definitely not be compromised if "the zero function" is replaced by "any function in $\mathscr{L}^2$ that is zero almost everywhere". Word heavy ... but not incorrect! I am trying to find out if there is any theorem/result that actually doesn't work in the pseudo-metric setting. $\endgroup$ – sami.spricht.sprache May 24 '17 at 19:08

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