Assume that a function $g$ has Cesaro averages $$ \lim_{x\to +\infty} \frac{1}{x}\int_0^x g(y) dy = \lim_{x\to +\infty} \frac{1}{x}\int_{-x}^0 g(y) dy =: \bar g \tag{1} $$ and $f$ is integrable.
Under which additional assumptions does it hold that $$ \int_{-\infty}^\infty f(x) g(ax) dx \to \bar g \cdot \int_{-\infty}^\infty f(x) dx, a\to+\infty? $$ I do not want to assume too much regularity for $f$ and/or $g$. (In my case $g$ is a sample of a special stationary ergodic process.)