That is, the Borel sets are the smallest & sigma;-algebra of subsets of " A " ? containing all the open sets.
42.
For some Borel set \ textstyle B, the namesake of this measure is revealed when the \ textstyle n \, th factorial moment measure reduces to:
43.
She had continued her research on Borel sets and in 1941 defended her thesis, but before she received her degree the family fled the advancing German troops.
44.
Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
45.
Halmos rather confusingly uses the term " Borel set " for elements of the ?-ring generated by compact sets, and defines Haar measure on these sets.
46.
The ?-algebra of Borel sets is most popular, but not the only choice ( Baire sets, universally measurable sets etc . are used sometimes ).
47.
The Borel sets are classified in the Borel hierarchy based on how many times the operations of complement and countable union are required to produce them from open sets.
48.
The measure that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.
49.
The measure that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.
50.
With no atoms, where \ textstyle B is a Borel set and \ textstyle f ^ {-1 } denotes the inverse of the function \ textstyle f.
