# Does $\mathcal{L}^2(\mathbb{R})$ form a metric space with this distance/similarity measure?

Consider the set $\mathcal{L}^2(\mathbb{R})$, where two functions $f$ and $g$ are said to be equal, if they agree almost everywhere. I would like to define a distance/similarity measure and would like to study its properties such as whether it would along with this set, forms a metric space.

Let $f$,$g$ be two functions on this set. We define a normalized cross correlation function as $$h(x) = \frac{\int \limits_{-\infty}^{\infty}f(t)g(t+x)\mathrm dt}{\int \limits_{-\infty}^{\infty}f(t)^2\mathrm dt \int \limits_{-\infty}^{\infty}g(t)^2\mathrm dt}$$

and our distance measure is given as $$d(f,g) = \frac{1}{\sup_x |h(x)|} - 1.$$

What I am interested to know is whether this set along with this distance measure qualifies as a metric space and also whether it has any interesting properties.

Modification (as it doesnt sound interesting in this current form) I hope its ok in this special case.

First we define an equivalence class and then modify the metric (infact would like to call it a similarity rather than metric and disacuss its properties and ask whether it would be modified to a metric if necessary)

Equivalence class :