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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 3

Question

Math 6210 • Real Analysis Spring 2020 • Question 3

Let \((X,\mathcal M,\mu )\) be a measure space with \(\mu (X)<\infty \). Denote by \(S\) the set of equivalence classes of measurable functions \(X\to \mathbb R\) where \(f\sim g\) if and only if \(f=g\) a.e. For \(f,g\in S\) define \[ d(f,g)=\int _X\frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}\,d\mu . \] Show that \((S,d)\) is a metric space and \(f_k\to f\) in \((S,d)\) if and only if \(f_k\to f\) in measure in \(X\).

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space with
\(\mu(X)<\infty\). Denote by \(S\) the set of equivalence classes of measurable
functions \(X\to\mathbb R\) where \(f\sim g\) if and only if \(f=g\) a.e. For
\(f,g\in S\) define
\[
d(f,g)=\int_X\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}\,d\mu.
\]
Show that \((S,d)\) is a metric space and \(f_k\to f\) in \((S,d)\) if and only if
\(f_k\to f\) in measure in \(X\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measurable Functions Measure Theory