Category Archives: Real Analysis

Weak Young’s convolution theorem

잘 알려진 Young’s convolution inequality는 다음과 같다. Theorem. For all such that     and , , we have and     이제 한 쪽의 integrability가 조금 약해져도 Young’s convolution theorem이 성립한다. Theorem. Let and satisfies     Then for any and , and there is a constant such that     여기서 이란 다음과… Read More »

Density extension

Proposition. Let and let be a linear (respectively, nonnegative sublinear) operator defined on a dense linear subspace of and taking values in . If     for all , then has a unique extension to a linear operator from to for which the inequality holds for all . Proof. First, we assume that is linear.… Read More »

Egorov’s Theorem

Theorem (Egorov). 이라 하고 를 sequence of measurable function이라 하고 a.e.라 하자. 그리고 가 finite a.e.라 하자. 임의의 에 대하여 이고 uniformly하게 하는 measurable set이 존재한다. Remark. 책에 따라 가정이 다르며, 설명이 조금 불충분한 편이다. 일반적으로 complete하지 않은 measure space는 sequence of measurable function이 almost everywhere convergence가 있다 할지라도 targetting function이 measurable하다는 보장을 할 수… Read More »

Lebesgue Outer measure and Continuity from below

Lebesgue outer measure인 경우에는 continuity from below가 measurability에 상관없이 항상 성립한다. Problem. If , then     Proof. If , then . So we may assume that . Let be a measurable hull of and define . Then note that is also a measurable hull of since for any . So . Also,    … Read More »

Arzela-Ascoli Theorem

1. Motivation of our main Theorem In metric space, topological compactness and sequential compactness are equivalent. In , we have a Heine-Borel property. However, the following example gives that closedness and boundedness do not gurantee the compactness in general. Example 1 This example is related to sequential compactness. Consider and where has 1 at th… Read More »