Is every subset of a Borel set Lebesgue measurable?

Question

Is it true that every subset of a Borel set is Lebesgue measurable? Why?

Answer 1

Any subset of $\mathbb R$ is a subset of a Borel set (namely $\mathbb R$). So any non-Lebesgue measurable set of reals is a counterexample.

Answer 2

Why would you expect this? The interval (0,1) is Borel, but not all of its subsets are Lebesgue-measurable (assuming Axiom of Choice).

Answer 3

No. Since $[0,1]$ is Borel and you can create a Vitali set on $[0,1]$.

Answer 4

No it is not true...As one can construct a non-measurable set in open interval (0,1)...