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+---
+title: Signed Measures
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 10
+---
+
+# {{ page.title }}
+
+{% definition Signed Measure %}
+A *signed measure* on a σ-algebra $\mathcal{A}$ on a set $X$
+is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that
+{: .mb-0 }
+
+- $\mu(\varnothing) = 0$,
+- either there is no $A \in \mathcal{A}$ with $\mu(A) = -\infty$
+  or there is no $A \in \mathcal{A}$ with $\mu(A) = \infty$,
+- for every sequence $(A_n)_{n \in \NN}$ of
+  pairwise disjoint sets $A_n \in \mathcal{A}$
+  {: .my-0 }
+  
+  $$
+  \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
+  = \sum_{n=0}^{\infty} \mu(A_n).
+  $$
+{% enddefinition %}
+
+{% definition Measure Space %}
+A *measure space* is a triple $(X,\mathcal{A},\mu)$ of
+a set $X$,
+a σ-algebra $\mathcal{A}$ on $X$
+and a measure $\mu$ on $\mathcal{A}$.
+{% enddefinition %}