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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2014 • Question 4

Question

Math 6210 • Real Analysis Fall 2014 • Question 4

Let \(f\in L^1(\lambda ,[0,1])\) have the property that for any measurable set \(A\) with \(\lambda (A)=\frac {1}{\pi }\) we have \(\int _A f\,d\lambda =0\). Show that \(f=0\) almost everywhere.

Raw LaTeX

Let \(f\in L^1(\lambda,[0,1])\) have the property that for any
measurable set \(A\) with \(\lambda(A)=\frac{1}{\pi}\) we have
\(\int_A f\,d\lambda=0\). Show that \(f=0\) almost everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lp Spaces and Sequence Spaces Measure Theory