It would be nice if average values of integrable functions obeyed the following rules on an interval [a, b].

a. av(ƒ + g) = av(ƒ) + av(g)

b. av(kƒ) = k av(ƒ) (any number k)

c. av(ƒ) ≤ av(g) if ƒ(x) ≤ g(x) on [a, b].

Do these rules ever hold? Give reasons for your answers.