In regression, the predictors $x$ is most often taken as given constants, not as random. The name design matrix comes from that, the statistician has chosen (designed) the $x$ values where the observations is taken. This viewpoint is especially relevant for design of experiments.

In counting process models, time dependence is important, and the sigma-algebras is used there. So you could expect, maybe, that the same notation could be useful in regression settings where time dependence is an issue. Sometimes, one uses lagged values of the response $y$ in modeling time dependence. Now, the response $y$ becomes part of the predictors $x$, and the supposition that $y$ is random while $x$ is deterministic becomes "problematic". And there is indeed a book, https://www.amazon.com/Regression-Models-Analysis-Benjamin-Kedem/dp/0471363553/ref=sr_1_1?ie=UTF8&qid=1535912891&sr=8-1&keywords=regression+models+for+time+series+analysis&dpID=41l1u-nkkGL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch "Regression Models for Time Series Analysis" (Wiley) that uses measure-theoretic notation.

In regression, the predictors $x$ is most often taken as given constants, not as random. The name design matrix comes from that, the statistician has chosen (designed) the $x$ values where the observations is taken. This viewpoint is especially relevant for design of experiments.

In counting process models, time dependence is important, and the sigma-algebras is used there. So you could expect, maybe, that the same notation could be useful in regression settings where time dependence is an issue. Sometimes, one uses lagged values of the response $y$ in modeling time dependence. Now, the response $y$ becomes part of the predictors $x$, and the supposition that $y$ is random while $x$ is deterministic becomes "problematic". 

And there is indeed a book "Regression Models for Time Series Analysis" (Wiley) that uses measure-theoretic notation.