The Borel sets in the statement of the theorem also refers to the ?-algebra generated by the " open " sets.
32.
For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective maps defined on Polish spaces.
33.
It can be proved that for every Borel set contained in, and for every, there exist a closed set and an open set such that
34.
The code for a lightface Borel set " A " can be used to inductively define a tree whose nodes are labeled by codes.
35.
For example, it can be shown that an inverse image f ^ {-1 } [ 0 ] is a non-Borel set.
36.
Alternatively, the number of points of " N " located in some Borel set " B " is often written as:
37.
The left Haar measure satisfies the inner regularity condition for all ?-finite Borel sets, but may not be inner regular for " all"
38.
For instance, a "'real-valued measurable function "'is a function for which the preimage of each Borel set is measurable.
39.
Given a countably additive measure ? on " X ", a measurable set is one that differs from a Borel set by a null set.
40.
Unfortunately, without the axiom of choice, it is not clear that the "-Borel sets " are " closed under wellordered union.
