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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 2

Question

Math 6210 • Real Analysis Fall 2008 • Question 2

Let \((\mathcal M,X,\mu )\) be a finite positive measure space.

(a) Show that \(L^2(X)\subset L^1(X)\), and that the inclusion \(i:L^2(X)\subset L^1(X)\) has norm \(\lVert i\rVert =\sqrt {\mu (X)}\).

(b) Let \(\mathcal M'\subset \mathcal M\) be the subset of measurable sets \(E\) with \(\mu (E)>0\). Show that if \[ L^1(X)=L^2(X), \] then \[ \inf _{E\in \mathcal M'}\mu (E)>0. \]

Raw LaTeX

Let \((\mathcal M,X,\mu)\) be a finite positive measure space.

(a) Show that \(L^2(X)\subset L^1(X)\), and that the inclusion
\(i:L^2(X)\subset L^1(X)\) has norm \(\lVert i\rVert=\sqrt{\mu(X)}\).

(b) Let \(\mathcal M'\subset \mathcal M\) be the subset of measurable sets \(E\)
with \(\mu(E)>0\). Show that if
\[
L^1(X)=L^2(X),
\]
then
\[
\inf_{E\in\mathcal M'}\mu(E)>0.
\]

Topic Tags

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