%0 Journal Article
%T On Some Topological Properties of Normed Boolean Algebras
%A Vesna Gotovac &#272
%A oga&#353
%J Advances in Pure Mathematics
%P 367-387
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.145021
%X This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>(</mo>
   <mrow>
    <mi>E</mi><mn>,</mn><mi>ℰ</mi><mn>,</mn><mi>μ</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>
 such that the N.B.A. is isomorphic to <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>(</mo>
   <mrow>
    <mover accent='true'>
     <mi>ℰ</mi>
     <mo>˜</mo>
    </mover>
    <mn>,</mn><mover accent='true'>
     <mi>μ</mi>
     <mo>˜</mo>
    </mover>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>
 resulting from the factorisation of initial <i>σ</i>-algebra by the ideal of negligible sets. We focus on topological properties <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>(</mo>
   <mrow>
    <mover accent='true'>
     <mi>ℰ</mi>
     <mo>˜</mo>
    </mover>
    <mn>,</mn><mover accent='true'>
     <mi>μ</mi>
     <mo>˜</mo>
    </mover>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>
 in general setting when <i>μ</i> can be an infinite measure. In case when <i>μ</i><i> </i>is infinite, we also consider properties of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mover accent='true'>
    <mi>ℰ</mi>
    <mo>˜</mo>
   </mover>
   <mrow>
%K Compact
%K Locally Compact
%K Polish Space
%K Separable
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=133389