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  "body": "\n## Definition\n\nLet $X$ be a topological space.\nThe Borel σ-algebra $\\mathcal{B}(X)$ is the smallest [[σ-algebra|04WSW8T3SY]] on $X$ that contains each open subset of $X$. It is the intersection of all σ-algebras on $X$ that contain the open sets.\nA Borel set of $X$ is an element of $\\mathcal{B}(X)$.\n\n## Remarks\n\nOn the [[real line|08J2M1JYAG]], $\\mathcal{B}(\\mathbb{R})$ is also the σ-algebra generated by the open [[intervals|0TK8Y3J5RW]] $(a, b)$. The half-lines $(-\\infty, b]$ with $b \\in \\mathbb{R}$ also generate it.\nEach open set, each closed set, each countable set and each interval of $\\mathbb{R}$ is a Borel set.\nSome subsets of $\\mathbb{R}$ are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.\n\n## Explanations\n\n### L1 human\n\nWe want to give a probability to intervals such as \"the waiting time is between $2$ and $5$ minutes\".\nThe Borel sets are all the sets that we can build from intervals with complements and countable unions, again and again. So they are the events that come from intervals.\n",
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