In probability theory, an isotropic measure is any mathematical measure that is invariant under linear isometries. It is a standard simplification and assumption used in probability theory. Generally, it is used in the context of measure theory on n {\displaystyle n} -dimensional Euclidean space, for which it can be intuitive to study measures that are unchanged by rotations and translations. An obvious example of such a measure is the standard way of assigning a measure to subsets of n-dimensional Euclidean space: Lebesgue measure.
Definition An isotropic measure on R d {\displaystyle \mathbb {R} ^{d}} is a (Borel) measure that is absolutely continuous on R d ∖ { 0 } {\displaystyle \mathbb {R} ^{d}\smallsetminus \{0\}} and that is invariant under linear isometries of R d {\displaystyle \mathbb {R} ^{d}} . Alternatively, an isotropic measure, μ ( d z ) {\displaystyle \mu (dz)} , is a measure for which there exists a real density function μ 0 ( r ) {\displaystyle \mu _{0}(r)} on ( 0 , ∞ ) {\displaystyle (0,\infty )} such that μ ( d z ) = μ 0 ( | z | ) d z {\displaystyle \mu (dz)=\mu _{0}\left(|z|\right)dz} for z ≠ 0 {\displaystyle z\neq 0} .
Example The Lebesgue measure on R d {\displaystyle \mathbb {R} ^{d}} is invariant under linear isometries and is hence an isotropic measure. In this case, μ ( d z ) = d z {\displaystyle \mu (dz)=dz} . For d = 1 {\displaystyle d=1} , the linear isometries of R 1 {\displaystyle \mathbb {R} ^{1}} are of the form f ( x ) = x + c {\displaystyle f(x)=x+c} or f ( x ) = − x + c {\displaystyle f(x)=-x+c} , for some constant c ∈ R {\displaystyle c\in \mathbb {R} } . Hence an isotropic measure on R 1 {\displaystyle \mathbb {R} ^{1}} must satisfy μ ( A ) = μ ( − A + b ) {\displaystyle \mu (A)=\mu (-A+b)} , for any A ⊆ R 1 {\displaystyle A\subseteq \mathbb {R} ^{1}} and b ∈ R {\displaystyle b\in \mathbb {R} } . The measure μ ( d z ) = | z | − 2 d z {\displaystyle \mu (dz)=|z|^{-2}dz} , for z ≠ 0 {\displaystyle z\neq 0} , is one such isotropic measure.
Unimodal measure In probability theory it is common that another assumption is added to measures in addition to the measure being isotropic. A unimodal measure (or isotropic unimodal measure) is any isotropic measure μ ( d z ) = μ 0 ( | z | ) d z {\displaystyle \mu (dz)=\mu _{0}\left(|z|\right)dz} such that μ 0 ( r ) {\displaystyle \mu _{0}(r)} is nonincreasing on ( 0 , ∞ ) {\displaystyle (0,\infty )} . It is possible that μ ( { 0 } ) > 0 {\displaystyle \mu \left(\left\{0\right\}\right)>0} .
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