What is the role of $x^p$ for defining the Lebesgue Integral?

Question

We define the Lebesgue Space by $L^p(A)$={$f \colon A\to R$ such that $\int|f|^pd\nu<\infty$}
I was reading one paper on orlicz spaces and there they have written that "For generalization of the function $x^p$ entering in the definition of Lebesgue's space is replaced by a more general convex function $\phi$.

Orlicz space is defined by $L^{\phi}$={$f \colon A\to R$ such that $\int\phi(|kf(x)|d\nu<\infty$ for some k>0}

My question is what is the role of $x^p$ for defining the Lebesgue space? Or am I missing something?

Answer 1

If you take the example $\phi(x) = x^p$, where $1\le p < \infty$, then $L^\phi = L^p$. If you try to generalize $L^p$ by writing $$ \left\{f:A\to R \;\big|\;\int \phi(|f(x)|)\;d\nu < \infty\right\} $$ then in general this is not a linear space. The definition quoted of $L^\phi$ (corrected so that "for some $k>0$" is inside the bracket $\}$) is a linear space.