I am studying the book "Real Analysis" by Folland, and I have a question about the following. Folland writes on pg 37 that:
Since the collection of open intervals is invariant under translations and dilations, the same is true of Borel sets in $R$
I understand the "since $\dots$ dilations" part, but why does this mean that the same is true of Borel sets? It seems true, but what is the proof of this claim?
Best Answer
Proofs of things about $\sigma$-algebras often begin by saying "Let $A=$..." and proceed by showing that $A$ is a $\sigma$-algebra.
Here: Let $A$ be the collection of all sets $E$ such that every translate of $E$ is a Borel set and every dilate of $E$ is a Borel set. Show that $A$ is a $\sigma$_algebra. Since open intervals are in $A$ and the Borel sets are the smallest $\sigma$-algebra containing the open intervals it follows that every Borel set is in $A$.