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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2024 • Question 5

Question

Math 6210 • Real Analysis Spring 2024 • Question 5

We say \(f_1,\ldots \) is \(\lambda \)-Cauchy if for every \(\epsilon >0\) there exists \(N\) so that for \(i,j>N\) we have

\[ \lambda (\{x: |f_i(x)-f_j(x)|>\epsilon \})<\epsilon . \]

Show that if \(f_1,\ldots \in L^1(\lambda )\) is \(\lambda \)-Cauchy then there exists \(g\in L^1(\lambda )\) so that \(f_1,f_2,\ldots \) converges in measure to \(g\).

Raw LaTeX

We say \(f_1,\ldots\) is \(\lambda\)-Cauchy if for every \(\epsilon>0\) there exists \(N\) so that for
\(i,j>N\) we have

\[
\lambda(\{x: |f_i(x)-f_j(x)|>\epsilon\})<\epsilon.
\]


Show that if \(f_1,\ldots\in L^1(\lambda)\) is \(\lambda\)-Cauchy then there exists \(g\in L^1(\lambda)\) so
that \(f_1,f_2,\ldots\) converges in measure to \(g\).

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Lp Spaces and Sequence Spaces Measure Theory