reference request: Borel sigma-algebra of a subspace is the trace sigma-algebra

Question

Let $\tau$ be the standard metric topology on $\mathbb{R}$. For $E\subseteq\mathbb{R}$, let $$\tau_E=E\cap\tau:=\{E\cap U:U\in\tau\}$$ denote the subspace topology on $E$. We write $\sigma(\tau)$ and $\sigma(\tau_E)$ for the respective Borel $\sigma$-algebras on $\tau$ and $\tau_E$.

According to ProofWiki, we have the identity $$\sigma(\tau_E)=E\cap\sigma(\tau):=\{E\cap B:B\in\sigma(\tau)\}.$$

My question is this: Can we find a more authoritative reference for the above identity than proofwiki?

I need to reference this fact in a paper I'm about to submit, and I'm pretty sure proofwiki won't go over very well, lol!

Answer 1

Proofwiki itself cites "2005: René L. Schilling: Measures, Integrals and Martingales §3: Problem 10 (ii)":

while not the same notation and not a full solution as in Proofwiki, it is indeed this result.