Space of measurable function

Question

Let $X=\{f:[0,1]\to \mathbb{F}, f ~measurable~function,\int_{0}^{1} |f(x)|^xdx<\infty\}$ be a set with metric $\rho(f,g)=\int_{0}^{1} |f(x)-g(x)|^xdx$. It looks like $L^p[0,1], 0<p<1$ space, but it is not. Exist any name of this space? Thanks

Answer 1

These are called Variable Exponent Lebesgue Spaces. The link is to pdf notes by Cruz Uribe.