Perfect measure

A concept introduced by B.V. Gnedenko and A.N. Kolmogorov in [1] with the aim of "attaining a full harmony between abstract measure theory and measure theory in metric spaces" . The subsequent development of the theory has revealed other aspects of the value of this concept. On the one hand the class of perfect measures is very wide, and on the other, a number of unpleasant technical complications that occur in general measure theory do not arise if one restricts to perfect measures.

A finite measure $\mu$ on a $\sigma$- algebra $S$ of subsets of a set $X$ is called perfect if for any real-valued measurable function $f$ on $X$ and any set $E \subset \mathbf R$ such that $f ^ { - 1 } ( E) \in S$,

$$\mu ( f ^ { - 1 } ( E) ) = \inf \ \{ {\mu ( f ^ { - 1 } ( G) ) } : {G \supset E , G \in \mathfrak G } \} ,$$

where $\mathfrak G$ is the class of open subsets of $\mathbf R$. For $\mu$ to be perfect, it is necessary that for any real-valued measurable function $f$ on $X$ there exists a Borel set $B \subset f( X) \subset \mathbf R$ such that $\mu ( f ^ { - 1 } ( B) ) = \mu ( X)$, and sufficient that for any real-valued measurable function $f$ on $X$ and any set $E \subset \mathbf R$ for which $f ^ { - 1 } ( E) \in S$ there exists a Borel set $B \subset E$ such that

$$\mu ( f ^ { - 1 } ( E) ) = \mu ( f ^ { - 1 } ( B) ) .$$

Every discrete measure is perfect. A measure defined on a $\sigma$- algebra of subsets of a separable metric space that contains all open sets is perfect if and only if the measure of any measurable set is the least upper bound of the measures of its compact subsets. The restriction of a perfect measure $\mu$ defined on $S$ to any $\sigma$- subalgebra of $S$ is perfect. A measure induced by a perfect measure $\mu$ on any subset $X _ {1} \in S$ with $\mu ( X _ {1} ) > 0$ is perfect. The image of a perfect measure $\mu$ under a measurable mapping of $( X , S )$ into another measurable space is perfect. A measure is perfect if and only if its completion is perfect. For every measure on any $\sigma$- subalgebra of a $\sigma$- algebra $S$ of subsets of a set $X$ to be perfect it is necessary and sufficient that for any real-valued $S$- measurable function $f$ the set $f ( X)$ is universally measurable (that is, it belongs to the domain of definition of the completion of every Borel measure on $\mathbf R$). If $X \subset \mathbf R$ and if $S$ is the $\sigma$- algebra of Borel subsets of $X$, then every measure on $S$ is perfect if and only if $X$ is universally measurable.

Every space $( X , S , \mu )$ with a perfect measure such that $S$ has a countable numbers of generators $\{ S _ {i} \}$ separating points of $X$( that is, for all $x , y \in X$, $x \neq y$, there is an $i$: $x \in S _ {i}$, $y\notin S _ {i}$ or $x \notin S _ {i}$, $y \in S _ {i}$) is almost isomorphic to some space $( L , {\mathcal L} , \lambda )$, consisting of the Lebesgue measure on a finite interval and of a countable sequence (possibly empty) of points of positive mass (i.e., there is an $N \in S$ with $\mu ( N) = 0$ and a one-to-one mapping $\phi$ of $X \setminus N$ onto $L$ such that $\phi$ and $\phi ^ {-} 1$ are measurable and $\lambda = \mu \phi ^ {-} 1$).

Let $I$ be any index set and let $( X _ {i} , S _ {i} , \mu _ {i} )$ be a given space with a perfect measure for each $i \in I$. Put $X = \prod _ {i \in I } X _ {i}$ and let ${\mathcal A}$ be the algebra generated by the class of sets of the form $\{ {x \in X } : {x _ {i} \in A \in S _ {i} } \}$. If $\mu ^ \prime$ is a finitely-additive measure on ${\mathcal A}$ such that $\mu ^ \prime ( \{ {x \in S } : {x _ {i} \in A } \} ) = \mu _ {i} ( A)$ for all $i \in I$ and $A \in S _ {i}$, then: 1) $\mu ^ \prime$ is countably additive on $A$; and 2) the extension $\mu$ of $\mu ^ \prime$ to the $\sigma$- algebra $S$ generated by ${\mathcal A}$ is perfect.

Let $( X , S , P )$ be a space with a perfect probability measure and let $S _ {1}$, $S _ {2}$ be two $\sigma$- subalgebras of the $\sigma$- algebra $S$, where $S _ {1}$ has a countable number of generators. Then there is a regular conditional probability on $S _ {1}$ given $S _ {2}$, i.e. there is a function $p ( \cdot , \cdot )$ on $X \times S _ {1}$ such that: 1) for a fixed $x$, $p ( x , \cdot )$ is a probability measure on $S _ {1}$; 2) for a fixed $E$, $p ( \cdot , E )$ is measurable with respect to $S _ {2}$; and 3) $\int _ {F} p ( x , E ) P ( d x ) = P ( E \cap F )$ for all $E \in S _ {1}$ and $F \in S _ {2}$. Moreover, the function $p ( \cdot , \cdot )$ can be chosen in such a way that the measures $p ( x , \cdot )$ are perfect. Let $( X , S )$, $( Y , {\mathcal T} )$ be two measurable spaces and let $q ( \cdot , \cdot )$ be a transition probability on $X \times {\mathcal T}$, that is, $q ( \cdot , E )$ is measurable with respect to $S$ and $q ( x , \cdot )$ is a probability measure on ${\mathcal T}$ for all $x \in X$, $E \in {\mathcal T}$. If the $q ( x , \cdot )$ are discrete and $P$ is a perfect probability measure on $S$, then the measure $\int q ( x , \cdot ) P ( d x )$ is perfect.

Perfect measures are closely connected with compact measures. A class ${\mathcal K}$ of subsets of $X$ is called compact if $K _ {i} \in {\mathcal K}$, $i = 1 , 2 \dots$ and $\cap _ {i = 1 } ^ \infty K _ {i} = \emptyset$ implies that $\cap _ {i=} 1 ^ {n} K _ {i} = \emptyset$ for some $n$. A finite measure $\mu$ on $( X , S )$ is called compact if there is a compact class ${\mathcal K}$ such that for all $\epsilon > 0$ and $E \in S$ one can choose a $K \in {\mathcal K}$ and an $E _ {1} \in S$ such that $E _ {1} \subset K \subset E$ and $\mu ( E \setminus E _ {1} ) < \epsilon$. Every compact measure is perfect. For a measure to be perfect it is necessary and sufficient that its restriction to any $\sigma$- subalgebra with a countable number of generators be compact.

References

 [1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) [2] E. Marczewski, "On compact measures" Fund. Math. , 40 (1953) pp. 113–124 [3] C. Ryll-Nardzewski, "On quasi-compact measures" Fund. Math. , 40 (1953) pp. 125–130 [4] V.V. Sazonov, "On perfect measures" Transl. Amer. Math. Soc. (2) , 48 (1965) pp. 229–254 Izv. Akad. Nauk SSSR Ser. Mat. , 26 (1962) pp. 391–414 [5] D. Ramachandran, "Perfect measures" , 1–2 , Macmillan (1979)