How to transform the $L^2$ inner product on a sphere $T$ into an inner product on the unit sphere $S^2$ where $\alpha T = S^2$. for $\alpha>0$?

Question

Let $S^2$ be the unit sphere in $\mathbb{R}^3$ and let $T$ be another sphere such that $\alpha T = S^2$, for some $\alpha >0$. Now consider the $L^2$ inner product over the surface of the sphere $T$, $$ (f,g)_{L^2(T)} = \int_T f(x) \overline{g(x)} dx. $$ How I can I transform this inner product to the unit sphere? That is what is $$ (f,g)_{L(S^2)}, $$ in terms of $(f,g)_{L^2(T)}$, or equivalently what is $(f,g)_{L^2(T)}$ in terms of $(f,g)_{L(S^2)}$?