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Prove some properties for weighted averages of functions

by RoRi

With reference to the previous exercise which of the following properties are valid for weighted averages of a function on an interval [a,b]. Denote the weighted average of f with a weight function w on [a,b] by A(f).

  1. Additive property: A(f+g) = A(f) + A(g).
  2. Homogeneous property: A(cf) = cA(f) for all c \in \mathbb{R}.
  3. Monotone property: A (f) \leq A(g) if f \leq g on [a,b].

All of these properties are valid for weighted averages.

  1. Proof. We compute,

        \begin{align*} A(f+g) &= \frac{\int_a^b w(x)(f(x) + g(x))\, dx}{\int_a^b w(x) \, dx} \\ &= \frac{\int_a^b w(x)f(x) \, dx + \int_a^b w(x) g(x) \, dx}{\int_a^b w(x) \, dx} & (\text{Add. property of integral})\\ &= \frac{\int_a^b w(x) f(x) \, dx}{\int_a^b w(x) \, dx} + \frac{\int_a^b w(x)g(x) \, dx }{ \int_a^b w(x) \, dx} \\ &= A(f) + A(g). \qquad \blacksquare \end{align*}

  2. Proof. We compute,

        \begin{align*} A(cf) &= \frac{\int_a^b c f(x) w(x) \, dx}{\int_a^b w(x) \, dx} \\ &= c \cdot \frac{\int_a^b w(x) f(x) \, dx}{\int_a^b w(x) \, dx} & (\text{Homogeneous prop. of integrals})\\ &= c A(f). \qquad \blacksquare \end{align*}

  3. Proof. Assume f \leq g on [a,b], then since w is nonnegative (definition of a weight function) we have w(x) f(x) \leq w(x) g(x) for all x \in [a,b]. Next, by the monotone property of the integral we have

        \[ \int_a^b w(x) f(x) \, dx \leq \int_a^b w(x) g(x) \, dx.\]

    Then, since w is nonnegative, \int_a^b w(x) \, dx is also nonnegative and we have

        \begin{align*} \frac{\int_a^b w(x) f(x) \, dx}{\int_a^b w(x) \, dx} &\leq \frac{\int_a^b w(x) g(x) \, dx}{\int_a^b w(x) \, dx} \\ \implies A(f) &\leq A(g). \qquad \blacksquare \end{align*}

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