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- I am reading Folland's text on real analysis. He defines the Borel ##\sigma##-algebra on the extended reals and says this coincides with the usual definition of the Borel ##\sigma##-algebra being generated by open sets. I don't see how they coincide.
On page 45 in Folland's text on real analysis, he writes that we define Borel sets in ##\overline{\mathbb R}## by ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}##. Then he remarks that this coincides with the usual definition of the Borel ##\sigma##-algebra if we make ##\overline{\mathbb R}## into a metric space with metric ##d(x, y) = |\arctan x - \arctan y|##.
I don't see how the two coincide and would be grateful if someone could explain in more detail. I see how ##d## is a metric on ##\overline{\mathbb R}##, where e.g. ##\arctan(\infty)## should evaluate to ##\pi/2##, but I don't see how ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}## coincides with the usual definition of ##\mathcal B_{\overline{\mathbb R}}## being generated by the open sets in ##{\overline{\mathbb R}}##. In other words, how to show $$\sigma(\{\text{open sets in }\overline{\mathbb R}\})=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}?$$
I don't see how the two coincide and would be grateful if someone could explain in more detail. I see how ##d## is a metric on ##\overline{\mathbb R}##, where e.g. ##\arctan(\infty)## should evaluate to ##\pi/2##, but I don't see how ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}## coincides with the usual definition of ##\mathcal B_{\overline{\mathbb R}}## being generated by the open sets in ##{\overline{\mathbb R}}##. In other words, how to show $$\sigma(\{\text{open sets in }\overline{\mathbb R}\})=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}?$$