---- > [!definition] Definition. ([[simple function]]) > A function is called **simple** if it takes on finitely many values. > For $(X, \Sigma)$ and $(Y, \mathcal{T})$ [[σ-algebra|measurable spaces]], where adding\scaling makes sense in $Y$, and $f:X \to Y$ a [[measurable function]] with image $\{ c_{1},\dots,c_{n} \}$ one has $f=c_{1} \chi_{E_{1}}+\dots+c_{n} \chi _{E_{n}}$ where $E_{k}=f ^{-1}(\{ c_{k} \})$. Thus this function $f$ is a [[measurable function|measurable function]] if and only if $E_{1},\dots,E_{n} \in \Sigma$. ---- #### ---- #### 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 > ```