Top Page > Top Page of the English Version > Mathematics > Notes

Notes

These notes originate from my talks in informal seminars. They are just summaries of (well-)known facts without any original results.

1. Properties of G_δ subsets of R.  PDF file.
   It is well-known that there exists a function from the real line to itself that is continuous at irrational numbers and discontinuous at rational numbers. In contrast, there is no function that is continuous at rational numbers and discontinuous at irrational numbers. We prove this by examining the properties of G_δ subsets of the real line.
2. The Erdős-Sierpiński Duality Theorem.  PDF file.
   The two σ-ideals on the real line consisting of all null sets and of all meagre sets are similar in many respects. The Erdős-Sierpiński Duality Theorem, which explains many of such similarities, asserts that on the assumption of the continuum hypothesis, there exists an involution from the real line to itself whose image of a subset is meagre (resp. null) if and only if the original subset is null (resp. meagre).
3. Cauchy Functional Equation  PDF file.
   We prove that every Lebesgue measurable function satisfying the Cauchy functional equation is linear.