| Summary: | In this article, some prescriptions to define a distribution on the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math></inline-formula> of all rational numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers <i>asymptotically equiprobable</i> in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula>—the probability allotted to every single <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> asymptotically vanishes, while that of the subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> falling in an interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>⊆</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> goes to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></semantics></math></inline-formula>. We finally present some hints to complete sequencing without repeating the numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> as a prerequisite to laying down more distributions on it.
|
|---|