Lebesgue generalizations of Hilbert spaces? Is an L[p] space a generalization of Hilbert spaces using Lebesgue integration?
And if this is the case, is it true that Holder's and Minkowski's Inequalities are generalizations of the Cauchy-Schwarz and triangle inequalities using Lebesgue measures?
 A: No. To start, $L^p$ is a true Banach space, not a "generalized" Banach space.
I assume you use the term Lebesgue integration as contrasted with Riemann integration. You could, if you wanted, define the normed space of all functions $f:[0,1]\to\mathbf{R}$ with $|f|^p$ Riemann-integrable, with norm $\|f\|_p = \int |f|^p$. You might then hope to prove that this is a Banach space, but you would fail (the completion of this space is precisely $L^p([0,1])$).
If you did carry out this programme, you would still want to prove Minkowski's inequality and Holder's inequality, which are generalisations of the triangle and Cauchy-Schwarz inequalities to $p\neq 2$, even though you've decided that you hate Lebesgue.
Edit: In response to your new phrasing, I think it's best to say that Banach spaces are generalized Hilbert spaces. Note that $L^2([0,1])$ is a Hilbert space, but definitely requires Lebesgue integration to ensure that the space is complete. (In general, the correct way to view Lebesgue integration is as the "completion" of Riemann integration.)
A: $L^p$ space is a generalization of $L^2$ space, and if $p\neq2$ they are not Hilbert. Hilbert space is an abstract concept, and it is more general than $L^2$ spaces. $L^2$ is just one concrete example of a Hilbert space.
