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.