There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable.
In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.
Moreover, some authors add restrictions on the topological space that Baire sets are defined on, and only define Baire sets on spaces that are compact Hausdorff, or locally compact Hausdorff, or σ-compact.
7.1) gives an equivalent definition and defines Baire sets of a topological space to be elements of the smallest σ-algebra such that all continuous real-valued functions are measurable.
One reason for working with compact Gδ sets rather than closed Gδ sets is that Baire measures are then automatically regular (Halmos 1950, theorem G page 228).
Alternatively, Baire sets form the smallest σ-algebra such that all continuous functions of compact support are measurable (at least on locally compact Hausdorff spaces; on general topological spaces these two conditions need not be equivalent).
For locally compact Hausdorff topological spaces that are not σ-compact the three definitions above need not be equivalent.
[5] The Kolmogorov extension theorem states that every consistent collection of finite-dimensional probability distributions leads to a Baire measure on the space of functions.