Let $g(t)$ be a positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}$ (ie: $L^0(\mathbb{R})$).

I tried working in a weighted $L^2_w(\mathbb{R})$ space but these are not orthogonal so I can't use that type of argument. How could I go about showing this?

In a nutshel I'm trying to show it's a Schauder basis.