I am stuck in the following exercise in Functional Analysis for Probability and Stochastic Processes. An Introduction, 1.6, The cauchy equation:
Let $y:\mathbb{R}^{+}\to \mathbb{R}$ be a measurable function such that $x(s):=\lim_{t \to \infty}\frac{y(t+s)}{y(s)}$ exists for all $s>0$. Show that we must have $x(s)=e^{as}$ for some real $a$.
The main results about Cauchy-equation is $f(x+y)=f(x)+f(y)$ holds for $\mathbb{Q}$ implies $f$ is linear on $\mathbb{Q}$ and continuous if it's measurable.
My idea is doubly using cauchy equation: let $$ f(x):=\ln \circ x(s)=\lim_{t \to \infty} \ln y(t+s) -\ln (s) $$
Which is cauchy and in suffices we show that $\ln \circ y$ is also cauchy and thus $y=e^{as}$ for some $a$. However, limit of which does not exist. I also find a counterexample that $y=\frac{1}{s}$, then $x$ should be $0$, is $a$ there are allow to be $-\infty$?
Any help would be highly appreciated.