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May
11
comment Is the empty set Lebesgue measurable?
To be honest, I find the result quite counter-intuitive. For example, if $S$ and $T$ are distinct measurable non-empty sets of real numbers with $S$ a subset of $T$, then we can say that $\mu(S) < \mu(T)$. This is intuitive - if you have a line and then you cut a bit off then the resulting line is shorter. However if you remove the non-empty restriction then this breaks down - let $S = \emptyset$ and $T = \{0\}$, then $S$ is a distinct subset of $T$ but $\mu(S) = \mu(T)$. However, despite my intuition, the empty set is indeed considered to be measurable, as answers have indicated.
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answered Problems that are largely believed to be true, but are unresolved