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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2024 • Question 5

Question

Math 6210 • Real Analysis Fall 2024 • Question 5

Let \((X,\mathcal {M},\mu )\) be a measure space with \(\mu (X)<\infty \). Show that if \(1\le q< p\le \infty \) then \(L^q(X)\supseteq L^p(X)\). Moreover, if \(X\) contains an infinite family of pairwise disjoint subsets of positive measure, then \(L^q(X)\supsetneq L^p(X)\).

Raw LaTeX

Let \((X,\mathcal{M},\mu)\) be a measure space with \(\mu(X)<\infty\). Show that if \(1\le q<
p\le\infty\) then \(L^q(X)\supseteq L^p(X)\). Moreover, if \(X\) contains an infinite family
of pairwise disjoint subsets of positive measure, then \(L^q(X)\supsetneq L^p(X)\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory