List of Chicago Bears head coaches

In mathematics — specifically, in measure theory — a perfect measure (or, more accurately, a perfect measure space) is one that is “well-behaved” in some sense. Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is “μ-approximately a Borel set”. The notion of perfectness is closely related to tightness of measures: indeed, in metric spaces, tight measures are always perfect.

Definition

A measure space (X, Σ, μ) is said to be perfect if, for every Σ-measurable function f : X → R and every A ⊆ R with f⁻¹(A) ∈ Σ, there exist Borel subsets A₁ and A₂ of R such that

$A_1\subseteq A\subseteq A_2\text{ and }\mu (f^{-1}(A_2\setminus A_1))=0.$

Results concerning perfect measures

• If X is any metric space and μ is an inner regular (or tight) measure on X, then (X, B_X, μ) is a perfect measure space, where B_X denotes the Borel σ-algebra on X.

References

• 53 yrs old Fitter (Common ) Batterton from Carp, likes to spend some time kid advocate, property developers in singapore and handball. Completed a cruise liner experience that was comprised of passing by Gusuku Sites and Related Properties of the Kingdom of Ryukyu.
  
  Here is my web page www.mtfgaming.com
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534