Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1997 • Question 2

Question

Math 6210 • Real Analysis Fall 1997 • Question 2

Let \(0<p<r<q\). Show that if \((X,\mathfrak M,\mu )\) is a (positive) measure space, then \[ L^r(\mu )\subset L^p(\mu )+L^q(\mu ); \] more precisely, for every \(f\in L^r(\mu )\) there exists \(g\in L^p(\mu )\) and \(h\in L^q(\mu )\) such that \(f=g+h\).

Raw LaTeX

Let \(0<p<r<q\). Show that if \((X,\mathfrak M,\mu)\) is a (positive)
measure space, then
\[
L^r(\mu)\subset L^p(\mu)+L^q(\mu);
\]
more precisely, for every \(f\in L^r(\mu)\) there exists \(g\in L^p(\mu)\) and
\(h\in L^q(\mu)\) such that \(f=g+h\).

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Outer Measures and Caratheodory Measurability