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  "body": "\n## Summary\n\nThe product σ-algebra on a Cartesian product of [[measurable spaces|N4KDT83V3S]] is the smallest [[σ-algebra|0SV8K0TDGY]] that makes every coordinate projection measurable.\nFor two factors it is generated by the measurable rectangles $A \\times B$.\nIt is the σ-algebra on which [[product measures|77PV0ZAVXT]] live and on which Fubini's theorem is stated.\nFor infinite products it is generated by the cylinder sets, which constrain only finitely many coordinates.\n\n## Explanations\n\n### L1 human\n\nTo measure sets in a product $X \\times Y$, start from the rectangles $A \\times B$ built from measurable pieces of each factor.\nThe product σ-algebra contains these rectangles and everything one can build from them with countable unions, countable intersections and complements.\nFor the real line with its Borel sets, this produces exactly the Borel sets of the plane.\nFor infinitely many coin tosses, it contains the events that depend on finitely many tosses, and countable combinations of them, such as \"heads appears infinitely often\".\n",
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