# A Hypothesis on measurable sets

1. Apr 3, 2017

### zwierz

Let $D$ be a bounded subset of $\mathbb{R}^m$; equip the set $2^D$ with the Hausdorff distance. We have obtained a semimetric space. Turn this space into a metric space by the corresponding factorization. Denote this factor space by $F$. Let $\overline F$ stand for comleteion of $F$.

Hypothesis. The standard Lebesgue $\sigma-$ algebra is an of first Baire category set in $\overline F$.
(The inclusion $\sigma \subset \overline F$ is understood in sense of corresponding embeddings)

I tried to formalize an intuitive understanding (my understanding) that the set measurable subsets should be very small in the set of all subsets of $\mathbb{R}^m$. Is that true what do you think?

2. Apr 8, 2017

### PF_Help_Bot

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