Consider a $\mathfrak{B}(\mathbb{R})$ - Borel $\sigma$-algebra on $\mathbb{R}$. It's easy to see that $\#\mathfrak{B}(\mathbb{R}) = 2^{\aleph_0}$. And we now that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$ - Lebesgue $\sigma$-algebra. I want to show that there is exist such subset of real line , which are Borel non measurable and Lebesgue measurable. But I don't know how?
Actually , I know about some examples. But maybe it possible to show easier than constructing a counterexample? For example I thought about finding some Lebesgue measurable sets , which are bigger than $\mathfrak{B}(\mathbb{R})$. And I thought about Cantor set.