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# Prove some properties for weighted averages of functions

With reference to the previous exercise which of the following properties are valid for weighted averages of a function on an interval . Denote the weighted average of with a weight function on by .

1. Additive property: .
2. Homogeneous property: for all .
3. Monotone property: if on .

All of these properties are valid for weighted averages.

1. Proof. We compute, 2. Proof. We compute, 3. Proof. Assume on , then since is nonnegative (definition of a weight function) we have for all . Next, by the monotone property of the integral we have Then, since is nonnegative, is also nonnegative and we have 1. Andres says:

What if both functions have different weighted functions?

• Youssef says:

same question

• Awamoki says:

Doesn’t change the result because you could rewrite them even if there are different weighted functions using the properties of integrals