In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is an "open Baire set." What does this mean? Clearly they are not two separate adjectives, as $\{a<\varphi\}$ is clearly not always open.
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2$\begingroup$ Perhaps some background information would be helpful. An open set is not necessarily a Baire set (a set whose indicator function is a Baire function, or equivalently belongs to the Baire $\sigma$-algebra, generated by the zero sets). A compact Baire set is $G_\delta$, so in a compact Hausdorff space, the Baire open sets are the $F_\sigma$ open sets. Therefore if $X$ is uncountable, the complement of a point in the product space $2^X$ is a non-Baire open set. In a perfectly normal space, e.g. a metrizable space, every open set is a Baire set. $\endgroup$Robert Furber– Robert Furber2023-11-11 01:42:43 +00:00Commented Nov 11, 2023 at 1:42
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1$\begingroup$ Of course, every open set in every topological space has the Baire property and belongs to the Baire property $\sigma$-algebra, which is why we have to be careful to distinguish these notions, all arising from generalizing results in Réné Baire's thesis. $\endgroup$Robert Furber– Robert Furber2023-11-11 01:43:53 +00:00Commented Nov 11, 2023 at 1:43
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The function $\varphi$ in H.-J. is also assumed to be lower semi-continous, which is why $\{a<\varphi\}$ is open. This set is a Baire set because $\varphi$ is a Baire function.
