The law of large numbers and a functional equation

We deal with the linear functional equation (E) ${\displaystyle g\left(x\right)={\sum }^r\_\{i=1\}{p}_{i}g\left(c_{i}x\right)}$, where g:(0,∞) → (0,∞) is unknown, ${\displaystyle (p₁,...,p_r)}$ is a probability distribution, and ${\displaystyle c_i}$'s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli's Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.