Why should a physicist care about measurability of random variables? I'm working with some physicists at the moment, and I made the remark that random variables are, technically, defined as measurable functions on some "background" probability space, hence requiring the choice of two sets and two sigma algebras.   They immediately asked where the term "measurable" came from (I gave the brief history about measuring lengths and areas), and then asked if it had anything to do, in general, with whether or not something was "measurable in a laboratory" (I said I wasn't sure).  They then proceeded to say that it was nonsensical to consider any "non-measurable" random variables because such things are, almost by definition, not "interesting" to the physicist.

So, how can I explain to my physicist colleagues that we actually do care about measurability?  Is there an example of a physical quantity which would otherwise be interesting but which turns out to be non-measurable (in the mathematical sense), and which might justify my mathematical nitpickiness to them?  I'm OK with examples from ordinary probability or stochastic process theory.
This is easily explained 

Suppose there are three states of the world a, b c.
And suppose we receive a signal $s$ from $\{a,b,c\}$ to $\mathbb{R}$ such that $s(a)= s(b)&gt; s(c)$. 

A random variable is measurable on the algebra generated by $s$ is precisely those that can't distinguish between states $A$ and $b$. So measurability is what you can measure in a lab if what you are measuring depends exclusively on the data given by the signal generating the $\sigma$-algebra. 

When there is a finite number of states in a measure space $\Omega$ every algebra of events is defined by a signal $s\colon \Omega\to \mathbb{R}$. A a random variable $f\colon \Omega \to \mathbb{R}$ is measurable only if $f= g\circ s$ for some function $g\colon \mathbb{R} \to \mathbb{R}$. So measirability is precisely measurability based on available data. 

With an infinite state space, say $[0,1]$ an event $E\subset [0,1]$ is non-measurable if it is inconceivable for you to describe how you would distinguish in any conceivable laboratory if a state lies in $E$ or its compliment when the signal you receive is $s(a) = a$ for all $a\in [0,1]$. You cannot conceivably imagine an experiment that can answer this question.  

Of course, if the physicist wants to integrate a function that is not measurable; then there is no way of doing this for the reasons above.
Let me tell you a story.

Certain dynamic light scattering techniques characterize their samples by finding the underlying variance of the motion distribution. That is to say they collect realizations of a random process. From this they can get things like diffusion coefficients, etc. Problem here is that sometimes the underlying physics of the model might dictate a Cauchy distribution which, as we know doesn't have a variance. A the end of the day they're trying to make sense of what the meaning of their experimentally measured variance means when the underlying model doesn't have a variance term...