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    Riemann integrability of $g$ when $|g(x)-g(y)|\le|f(x)-f(y)|$ for integrable $f$

    P-FOAQM

    Area
    Complex Analysis
    Topics
    Integrals, Riemann Integrability
    Status
    draft

    Suppose \(f, g: [0, 1] \to \RR\) where \(f\) is Riemann integrable and for \(x, y\in [0, 1]\), \[ \abs{g(x) - g(y)} \leq \abs{f(x) - f(y)} .\]

    Prove that \(g\) is Riemann integrable.

    Solution

    Write \(U(f), L(f)\) for the upper and lower sums of \(f\), so for \(\Pi\) the collection of all partitions of \([0, 1]\), \[ U(f) \da \inf_{P\in \Pi} U(f, P) && U(f, P) \da \sum_{k=1}^n \sup_{x\in I_k}f(x) \cdot \mu(I_k) \\ L(f) \da \sup_{P\in \Pi} L(f, P) && L(f, P) \da \sum_{k=1}^n \inf_{x\in I_k} f(x) \cdot \mu(I_k) .\]

    Note that integrability of \(f\) is equivalent to \[ \forall \eps \exists P \text{ such that } U(f, P) - L(f, P) < \eps \\ \iff \sum_{k=1}^n \qty{ \sup_{x\in I_k} f(x) - \inf_{x\in I_k} f(x)} \mu(I_k) < \eps .\]

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