I'm playing around with a collection of subsets of $\mathbb{R}^n$ let's call them $X_i$. What I want to know is, is the following condition sufficient for some $X_i$ being measurable?
Almost all elements of $X_i$ are surrounded by a neighbourhood contained in $X_i$
This is motivated by the idea that non measurable sets are "fractally wiggly"
Assuming standard topology and measure on $\mathbb{R}^n$ but I'm also interested in minimal assumptions regarding topology and measure that keep this a sufficient condition for being measurable.