# Why dont we only consider complete measure spaces?

This question has been asked a few months ago, but in my opinion, it has not yet received a satisfying answer.

So I ask again: Why should one consider non-complete measure spaces instead of requiring every measure space to be complete per definition?

Most answers on similar questions are like:

• If $$f$$ is a measurable function, one wants its codomain to contain fewer measurable sets, so that one finds more measurable functions beside $$f$$.
• Lebesgue-Lebesgue-measurable functions are pretty useless because not even continous functions have to be L-L-measurable (but diffeomorphisms are).
• If $$\Omega$$ is a probability space and $$X:\Omega \to \mathbb{R}$$ a random variable (where $$\mathbb{R}$$ is endowed with any $$\sigma$$-Algebra), its distribution (pushforward measure) isnt necessarily a complete measure.

But I see a mistake in those thoughts. The category of measurable spaces (set + $$\sigma$$-Algebra) is confused with the category of measure spaces (set + $$\sigma$$-Algebra + measure). Of course, Lebesgue-Borel-measurable functions are important, because they are the subject of separably-valued integration (on $$\mathbb{R}^n$$). But for a Bochner-measurable function (a.e. pointwise limit of simple functions) that one wants to integrate, it is irrelevant if there is a measure on its codomain. The $$L_1$$-functor goes $$L_1 : \mathsf{MeasureSpaces} \times \mathsf{BanachSpaces} \to \mathsf{BanachSpaces} \ ,\ (X,A,\mu) , E \mapsto L_1(X,\mu,E)$$ so it may be wrong to think of any sigma-Algebra on the codomain of an integrable function.

Also, if $$(X,A,\mu)$$ is a measure space and $$(X,\bar A,\bar \mu)$$ is its canonical completion, a function $$f:X \to E$$ into a Banach space $$E$$ is Bochner-measurable with respect to $$\mu$$, if and only if it is Bochner-measurable with respect to $$\bar \mu$$. So, for Bochner integration theory, non-complete measure spaces are completely redundant.

And concerning the distribution argument: If one has a random variable $$X : \Omega \to \mathbb{R}$$, I see no reason to postulate a $$\sigma$$-Algebra on $$\mathbb{R}$$ and the measurablitiy of $$X$$. I think it to be more natural to view $$\mathbb{R}$$ as simply a set and consider every subset of $$\mathbb{R}$$ to be measurable, whose preimage is measurable. This yields a $$\sigma$$-Algebra, bears no needless abandonment of per se measure-accessible subsets and if $$\Omega$$ is complete, the pushforward measure will be too.

Here are two arguments. I don't think you are addressing them (maybe because I don't speak categories and don't know much about Bochner integration). I use $$(\Omega, \mathcal{A}, \mu)$$ for a probability space, and $$X$$ for a r.v.
First, regarding the "distribution argument". Suppose we observe and want to study a normal distribution on $$\mathbb{R}$$. The co-domain is fixed, while the abstract outcome space $$\Omega$$ and r.v. $$X$$ are arbitrary. We are free to choose any combination of $$X$$, $$(\Omega, \mathcal{A}, \mu)$$, and we don't care which, as long as the pushforward is the normal distribution. Our application is only concerned with $$\sigma$$-summability in $$\mathbb{R}$$, this is why we postulate a $$\sigma$$-algebra there. In my understanding, the only real purpose of having abstract outcome and event spaces ($$\Omega$$ and $$\mathcal{A}\subset 2^{\Omega}$$) is to formalise interactions between random variables when we have more than one. If we start with $$(\Omega, \mathcal{A}, \mu)$$, $$X_i$$, and consider pushforward $$\sigma$$-algebras, as you suggest in the last paragraph, we risk having several incompatible $$\sigma$$-algebras in $$\mathbb{R}$$. That is, questions like "Is $$X_1\in (a,b)$$ more likely than $$X_2\in (a,b)$$?" aren't even guaranteed to be well-posed.
Second (which is related to the first), we often want $$g \circ X$$ to be a random variable on the same domain and co-domain $$\sigma$$-algebras as $$X$$ for a wide range of transformations $$g$$ (when we build a regression model, for example). This motivates the second bullet point and possibly others.