# Question on Step X of Rudin's proof of the Riesz Representation Theorem

I am working through Rudin's proof of the Riesz Representation Theorem in his Real and Complex Analysis texbook. The statement of the theorem is as follows:

I am stuck on Step X. The proof of Step X is given by Rudin as:

My Question is about the sets $$E_i$$. Specifically, I don't understand Rudin's explanation of why these sets have to be Borel sets. I understand why they are disjoint, why their union is K, and why f is a Borel measurable function, but I don't see why this implies the sets $$E_i$$ are Borel sets. I understand that K is a Borel set because it's closed, and so if the set {$$x: y_{i-1} } were a Borel set, then $$E_i$$ would be a Borel set since it's the intersection of two Borel sets. My confusion is about how we know the set {$$x: y_{i-1} } is a Borel set. Any elucidation on this point would be much appreciated.

As $$f$$ is continuous the intersection on the left hand side is an intersection of a an open and a closed set, thus the intersection is a Borel set $$\{ x: y_{i-1}