The Borel Sets
The Borel sets are simply defined as the elements of the borel $\sigma-algebra$.
In topological space, the borel sets are sets in a topological space that can be
formed from open sets or equivalently from closed sets, through the operations of
countable unions,countable intersection, and relative complements.In topological space, the borel sets are sets in a topological space that can be
formed from open sets or equivalently from closed sets, through the operations of
the Borel sets are named after Emile Borel.
Borel $\sigma-algebra$
This is the smallest $\sigma-algebra$ containing all open subsets of $\mathbb{R}$ the real
line.
line.
Examples of the Borel Sets
(1) A subsets of a locally compact Hausdorff topological space is called a borel set,
if it belongs to the smallest $\sigma-ring$ containing all compact sets.
(2) the sets of all rational numbers in [0,1] is a borel subset of [0,1].
(3) the set of irrational numbers in [0,1] is a borel subset of [0,1].
if it belongs to the smallest $\sigma-ring$ containing all compact sets.
(2) the sets of all rational numbers in [0,1] is a borel subset of [0,1].
(3) the set of irrational numbers in [0,1] is a borel subset of [0,1].
counter example
Every irrational number has a unique representative by a continued fraction,
Assume A to be the sets of all irrational numbers that corresponds to sequences, $A=\{a_1,a_2,a_3,...\}$ with the following properties such that there exists an infinite subsequence $B=\{a_{k0},a_{k1},...\}$ such that each element is a divisor of the next. Hence A is not a Borel set.
Infact A is analytic and complete in the class of analytic sets.
Assume A to be the sets of all irrational numbers that corresponds to sequences, $A=\{a_1,a_2,a_3,...\}$ with the following properties such that there exists an infinite subsequence $B=\{a_{k0},a_{k1},...\}$ such that each element is a divisor of the next. Hence A is not a Borel set.
Infact A is analytic and complete in the class of analytic sets.
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4 Comments
Nyc 1
ReplyDeleteReally proud of u dear!!
ReplyDeleteKeep up the good work!
My question ❓
ReplyDeleteIs the closed set 0,1 a borel set!??
Of course all open or closed sets are borel sets,
ReplyDeletesince any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space.
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