Non-measurable subset of a null set.

Question

I am reading measure theory,and I am searching an example in which a measurable null set have a non-measurable subset because this is the reason that,s why we are studying about complete measure.Please If someone have any information about it share with me??? Thanks.

Answer 1

You didn't specify what measure you want to consider, so I'll pick one: Borel measure on the reals. The Cantor middle thirds set has zero Borel measure and it contains non-Borel measurable subsets. One way to prove this last claim is to make use of the fact that there are only $c$ many Borel subsets of the real line, while there are $2^c$ many subsets of the Cantor middle thirds set.

This is probably the example @Édes István Gergely had in mind in his comment (if asked to be more specific), by the way.

Answer 2

Take the devil staircase $\psi:[0,1]\rightarrow [0,1]$ and consider the function $x+\psi(x)$. This map is strictly increasing from $[0,1]$ to $[0,2]$, and thus it is an homeomorphism, and it maps the cantor set in a set of zero measure. Take a non measurable set in the image of the cantor set (which exists since its image through $x+\psi$ has measure $1$) and consider its preimage. It is therefore a non Borel set contained in a null set on the line. This answer your question completely.