Let X {\displaystyle X} be a nonempty set. An outer measure on X {\displaystyle X} is a function μ ∗ : 2 X → [ 0 , ∞ ] {\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]} such that (i) μ ∗ ( ∅ ) = 0 {\displaystyle \mu ^{*}(\emptyset )=0} (ii) μ ∗ ( A ) ≤ μ ∗ ( B ) {\displaystyle \mu ^{*}(A)\leq \mu ^{*}(B)} if A ⊆ B {\displaystyle A\subseteq B} (iii) μ ∗ ( ∪ j = 1 ∞ A j ) ≤ ∑ j = 1 ∞ μ ∗ ( A j ) . {\displaystyle \mu *\left(\cup _{j=1}^{\infty }A_{j}\right)\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).}
![{\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a514e94c721b4c864074ebea3cc863d66fa11ef2)
