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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1992 • Question 3

Question

Math 6210 • Real Analysis 1992 • Question 3

Let \((X,\mathcal M,\mu )\) be a measure space, where the measure \(\mu \) is positive. Suppose \(f,f_n\in L^1(X,\mu )\), \(f_n\ge 0\) for each \(n\), \(f_n\to f\) a.e. and \[ \int _X f_n\,d\mu \to \int _X f\,d\mu \] Prove that \(f_n\to f\) in \(L^1(X,\mu )\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real valued function \(g\), \(g^+=\max (g,0)\)).

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space, where the measure
\(\mu\) is positive. Suppose \(f,f_n\in L^1(X,\mu)\), \(f_n\ge 0\) for each \(n\),
\(f_n\to f\) a.e. and
\[
\int_X f_n\,d\mu\to \int_X f\,d\mu
\]
Prove that \(f_n\to f\) in \(L^1(X,\mu)\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real
valued function \(g\), \(g^+=\max(g,0)\)).

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Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lp Spaces and Sequence Spaces Measure Theory