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For my thesis, I need to define $L^p$ spaces but in the literature (Real and Complex Analysis by Rudin, Measure Theory and Integration by de Barra, ...) it is not defined as a set of equivalence classes by a quotient space construction. Do you know any literature where this is done explicitly?

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    I'm not sure what emphasis is being placed here on "a set of equivalence classes by a quotient space construction". The notion of two functions being equivalent by agreeing everywhere except on a set of measure zero is explicit in Rudin's *Real and Complex Variables*. Perhaps you have a notion of representing this equivalence by a "quotient space" construction, but I don't know that anyone thinks working directly with operations well-defined with respect to the equivalence classes is logically deficient. – 2017-02-07
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    @hardmath I mean something like the construction in equation (4.8) on page 116 of the lecture notes https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf. I did not want to cite lecture notes, thats why I was asking here. – 2017-02-07
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    Note that the line above (4.7) says simply: $$f\sim^\mu g \iff f=g \;\;\; \mu-\text{almost everywhere}$$ So the quotient space construction in those lecture notes simply invokes the equivalence relation (and follows it with a note about the norm mapping being well-defined on these and the convenience of omitting the explicit mentions of equivalence classes. – 2017-02-07
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    Many authors make the same construction and use the same notation ($L^p$ versus $\mathcal L^p$) to distinguish between the set of functions versus the quotient space. Cohn's [Measure Theory](https://www.amazon.com/Measure-Theory-Birkhäuser-Advanced-Lehrbücher/dp/1461469554) comes to mind as a book which treats this carefully (and is a very good book in general). – 2017-02-07

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As hardmath has pointed out, Rudin's treatment is surely adequate. But to your question: just glancing at some of the books I have on my desk at the moment, I see that Royden's Real Analysis and Bogachev's Measure Theory would make good recommendations.

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The approach taken by Rudin in Real and Complex Variables (2nd edition, at hand) is to introduce the equivalence relation "a.e. [$\mu$]" on measurable functions in Chapter 1 (page 28), Section "The Role Played by Sets of Measure Zero". He notes, for example, that transitivity of this relation "is a consequence of the fact that the union of two sets of measure $0$ has measure $0$."

He then develops a bit of the machinery that is needed later, including the notion of $\mu$-completion of a $\sigma$-algebra (Thm. 1.36) and the notion of convergence of a function sequence "almost everywhere".

It may appear from the opening passages of Chapter 3 (page 66), Section "The $L^p$-spaces", that he is about to omit any formal definition in terms of equivalence classes, e.g. when expressing the $p$-norm:

$$ ||f||_p = \left\{\int_X |f|^p d\mu \right\}^{1/p} $$

The previous (Chapter 1) discussion already details that the integral being well-defined on the "a.e. [$\mu$]" equivalence classes.

This apparent omission is merely a pedagogical pause. Explicit discussion then takes place on pages 68-69 of the importance of these equivalence classes to the definition of a metric $d(f,g)$ on this function space by taking "the distance between $f$ and $g$ to be $||f-g||_p$." Thus:

The only other property which $d$ should have to define a metric space is that $d(f,g) = 0$ should imply that $f=g$. In our present situation this need not be so; we have $d(f,g) = 0$ precisely when $f(x) = g(x)$ for almost all $x$.

More paragraphs explain about the equivalence classes forming a metric space with respect to the $p$-norm:

When $L^p(\mu)$ is regarded as a metric space, then the space which is really under consideration is therefore not a space whose elements are functions, but a space whose elements are equivalence classes of functions. For the sake of simplicity of language, it is, however, customary to relegate this distinction to the status of a tacit understanding and to continue to speak of $L^p(\mu)$ as a space of functions. We shall follow this custom.

There follows the book's first really important result (Thm. 3.11) about the $L^p(\mu)$ spaces, that these are a complete metric space, i.e. that every Cauchy sequence converges.

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    Indeed. I had only considered page 66 on its own because I was looking for a very short definition for the appendix for my thesis in statistics (for functional pca, you need L^2 Hilbert spaces). Thanks for the additional context you gave (+1). – 2017-02-07