---- - [ ] better to define in terms of unique translation invariance > [!definition] Definition. ([[Lebesgue measure]]) > Up to scaling, there is a unique translation-invariant [[locally finite]] [[measure]] $\lambda=\lambda_{n}$ on the [[Borel set|Borel σ-algebra]] $\mathcal{B}(\mathbb{R}^{n})$. Normalizing so that $\lambda([0,1]^{n})=1$, $\lambda$ is called the **Lebesgue measure on $\mathbb{R}^{n}$**. The existence of $\lambda$ is outlined as follows; first for $\mathbb{R}$, then inductively to $\mathbb{R}^{n}$. [[uniqueness theorem for Lebesgue measure|See uniqueness proof here.]] > > Though not countably additive on all of $2^{\mathbb{R}}$, the [[Lebesgue outer measure]] $|\cdot|$ on $\mathbb{R}$ *is* countably additive when restricted to the $\sigma$-algebra $\mathcal{L}(\mathbb{R})$ (called the **Lebesgue $\sigma$-algebra**) of sets that are almost [[Borel set|Borel]].[^1] We call these sets **Lebesgue measurable sets**, and we call the [[measure]] obtained by restricting $|\cdot|$ to $\mathcal{L}(\mathbb{R})$ resp. $\mathcal{B}(\mathbb{R})$ the **Lebesgue measure** on $\big( \mathbb{R}, \mathcal{L}(\mathbb{R}) \big)$ resp. $\big( \mathbb{R}, \mathcal{B}(\mathbb{R}) \big)$.[^2] > The proof proceeds by way of five lemmas. The first four form the bulk of the work, showing that $|\cdot|_{\mathcal{B}(\mathbb{R})}$ is additive. The fifth extends this result to show $|\cdot|_{\mathcal{L}(\mathbb{R})}$ is additive. >1. additivity of outer measure if one of the sets is open >2. additivity of outer measure if one of the sets is closed >3. approximation of Borel sets from below by closed sets >4. additivity of outer measure if one of the sets is a Borel set >5. additivity of outer measure for $\mathcal{L}(\mathbb{R})$ > The **Lebesgue measure $\lambda_{n}$ on $\mathbb{R}^{n}$** is defined inductively as the [[product measure]] $\lambda_{n}=\lambda_{n-1} \times \lambda_{1},$ where $\lambda_{1}$ is the Lebesgue measure on $\mathbb{R}$. (This makes sense in light of [[products and Borel σ-algebras commute for second-countable spaces]].) > [!equivalence] > - [[characterization of Lebesgue measurability]] ^equivalence > [!basicnonexample] Warning. > Even though the construction of the Lebesgue measure is informed by the [[standard topology on the real line|standard]] [[topological space|topology]] on $\mathbb{R}$, *measurability is not a topological invariant*. ^nonexample [^1]: That is, Borel up to a set of [[measure zero]]: $\mathcal{L}$ consists of the sets $A \subset \mathbb{R}$ for which there exists a Borel set $B \subset A$ such that $|A - B|=0$. Clearly $\mathcal{B}(\mathbb{R}) \subset \mathcal{L}$. Note that $\mathcal{L}(\mathbb{R})$ is obviously a $\sigma$-algebra: if $A$ is such that $|A-B|=0$ for some $B$, then $|(X-A)-(X-B)|=0$, and if $E_{1},E_{2},\dots$ are all such that $|E_{1}-B_{1}|=0, |E_{2}-B_{2}|=0,\dots$ then $|\bigcup_{k=1}^{\infty}E_{k}- \bigcup_{k=1}^{\infty}B_{k}| \leq \sum_{k=1}^{\infty}|E_{k}-B_{k}|=0$. [^2]: Whether the [[σ-algebra]] under consideration is $\mathcal{L}(\mathbb{R})$ or $\mathcal{B}(\mathbb{R})$ usually does not matter much, since sets of [[measure zero]] usually do not matter much. Because all sets that arise from the usual operations of analysis are Borel sets, it is easier and not unsafe to assume that "Lebesgue measure" means outer measure on the Borel sets, unless the context indicates otherwise. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` We will prove this by way of four lemmas: 1. [[Lebesgue measure#^proposition-1|additivity of outer measure if one of the sets is open]] 2. additivity of outer measure if one of the sets is closed 3. approximation of Borel sets from below by closed sets 4. additivity of outer measure if one of the sets is a Borel set > [!proposition] Lemma 1. (Additivity of outer measure if one of the sets is open) > Suppose $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is open. Then $|A \sqcup G|=|A|+|G|.$ ^proposition-1 We can assume $|G|<\infty$ because otherwise both $|A \sqcup G|$ and $|A|+|G|$ equal $\infty$. Subadditivity of [[outer measure]] implies $|A \cup G| \leq |A|+|G|$. Thus we only need to prove the other direction. The [[standard topology on the real line]] has as [[basis for a topology|basis]] [[topology generated by a basis|the]] [[open interval|open intervals]] and is [[second-countable space|second-countable]], hence $G=\bigsqcup_{n=1}^{\infty}I_{n}$ for some intervals $I_{n}$ which we may take to be disjoint.