# An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space

By sum of two sets I mean $$A+B := \{x+y:x \in A \quad y \in B\}$$, and there is a tip in a book of real analysis by Zhou Minqiang which says:

“If $$A,B$$ are Borel sets in $$\mathbb{R}^{n}$$, $$A+B$$ may not be a Borel set.”

I want to know some specific examples.(Maybe $$\mathbb{R}^{1}$$ ?)

• I heard that we can construct a closed set in $\mathbb{R}^{2}$ project to a non-Borel set. Is that true? – YOTAL Sep 22 at 8:21
• A closed set in $R^2$ is a countable union of compacts. So its projection is a countable union of compact sets. Therefore it is a Borel set. – juan Sep 22 at 11:24
• Yes. You are right. Maybe just a Borel set in $\mathbb{R}^{2}$ @juan – YOTAL Sep 22 at 11:53