Is every element contained in a smallest measurable set?

The answer is no, and here is one example: Let $X=\mathbb{R}$, and define a set $E \subseteq \mathbb{R}$ to be measurable if either

  • $0 \notin E$ and $E$ is countable, or
  • $0 \in E$ and the complement $E^c$ is countable.

Then it is easy to see that there is no smallest measurable set containing $x=0$.

Tags:

Analysis

General Topology

Measure Theory

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