Why is it that given $m >n > 0$ and a Lebesgue measurable set of measure $m$, it must have a Lebesgue measurable subset of size $n$?
I had the following idea in mind : Calling our set $A$, since it is Lebesgue measurable, we know $\lambda(A) = \inf\{\sum_{k} l(I_k) : I_k \mbox{ is an interval cover of A}\}$. However, this is a bit of a definition from outside, so we cannot manipulate this definition to get subsets of $A$.
At the same time, $\lambda(A)$ is also the supremum over the measure of all compact subsets of the set. By this logic, we can find a compact set with measure $ > 1 - \epsilon$ for any $\epsilon > 0$, but how about being equal to a certain number? It's not clear to me.
Assume $A\subset \mathbb{R}^d$ and define $f(\lambda)=\mathcal{L}^d(\{x=(x_1,...x_d)\in A \vert ~ x_1<\lambda\})$ for $\lambda\in \mathbb{R}$. Show that $f$ is a continuous function and use the intermediate value theorem.