---- > [!definition] Definition. ([[Lebesgue measurable function]]) > A **Lebesgue measurable function** is a [[measurable function]] $f:\big(A \subset \mathbb{R}, \mathcal{L}(A)\big) \to \big(\mathbb{R}, \mathcal{B}(\mathbb{R})\big)$. Here, $\mathcal{B}(\mathbb{R})$ denotes the [[Borel set|Borel σ-algebra]] on $\mathbb{R}$ and $\mathcal{L}(A)$ denotes the [[Lebesgue measure|Lebesgue σ-algebra]] on $A$ (i.e., $\mathcal{L}(A)=\{ A \cap E: E \in \mathcal{L}(\mathbb{R}) \}$). ^definition > [!basicproperties] > - [[every Lebesgue measurable function is almost Borel measurable]] ^properties ---- #### ---- #### 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 > ```