= Lebesgue measure
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{title2=$\lambda$}
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Lebesgue measure on the <real numbers> extends interval length: $\lambda([a,b])=b-a$ for $a\leq b$. Its restriction to the <Borel sigma-algebra> is the unique translation-invariant <sigma-finite measure> on that <sigma-algebra> normalized by $\lambda((0,1])=1$. The full <Lebesgue measure> is the <completion of a measure> of this restriction; its domain is the <Lebesgue measurable sets>. Translation invariance follows by comparing a translated <Borel measure> on half-open intervals and applying the <uniqueness theorem for measures>. For the construction and interval properties, see https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/1bf275c0667d3e903e4de22b9fb4481b_MIT18_102s21_lec8.pdf[MIT lecture 8].
Every <singleton> has <Lebesgue measure> zero: $\{x\}\subseteq[x-\varepsilon,x+\varepsilon]$ gives $0\leq\lambda(\{x\})\leq2\varepsilon$ for every $\varepsilon>0$. Enumerating the <countable set> $\mathbb Q\cap[0,1]$ and applying <countable additivity> yields
$$
\lambda(\mathbb Q\cap[0,1])=\sum_{n=1}^{\infty}\lambda(\{q_n\})=0,
\qquad\lambda([0,1])=1.
$$
Thus a <dense set> can be a <null set>, and a <union> of <null sets> indexed by an <uncountable set> can have positive <Lebesgue measure>. The distinction explains both the <Dirichlet function> example and why <almost everywhere> is weaker than everywhere.
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