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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1995 • Question 1

Question

Math 6210 • Real Analysis Fall 1995 • Question 1

Let \(\mu \) be a positive finite measure on \((X,\mathfrak M)\) and \(g\in L^1(\mu )\). Suppose there exists \(0<M<\infty \) such that \[ \left |\int gs\,d\mu \right |\le M\lVert s\rVert _p\qquad (1\le p<\infty ) \] for all simple measurable functions on \(X\). Then show that \(g\in L^q(\mu )\) where \(\frac {1}{p}+\frac {1}{q}=1\).

Raw LaTeX

Let \(\mu\) be a positive finite measure on \((X,\mathfrak M)\) and
\(g\in L^1(\mu)\). Suppose there exists \(0<M<\infty\) such that
\[
\left|\int gs\,d\mu\right|\le M\lVert s\rVert_p\qquad(1\le p<\infty)
\]
for all simple measurable functions on \(X\). Then show that \(g\in L^q(\mu)\)
where \(\frac{1}{p}+\frac{1}{q}=1\).

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measurable Functions Measure Theory