My question is pretty straight forward:
I have a definition for the p-times Lebesgue-integrable functions:
$$ \mathscr{L}^p := \{ f:f \text{ measurable and } ||f||_{p} := \bigg( \int_{\Bbb R}|f|^p d\lambda \bigg) ^{1/p}<\infty \}$$
Is it important, that we have $||f||_{p}$ in this definition or would $||f||^{p}_{p}$ also suffice?
This question arose, because our definition for $$ l^p := \{ (x_n)_n: \sum^{\infty}_{n=1}|x_n|^p < \infty \}$$ omits the superscript of $1/p$, which would be equivalent to the above definition, with $\lambda$ being the count-measure on the $\sigma$-Algebra $\mathfrak{P}(\Bbb N)$ and having $||f||^{p}_{p}$. If instead
$$ l^p := \{ (x_n)_n: (\sum^{\infty}_{n=1}|x_n|^p)^{1/p} < \infty \}$$ then that would be equivalent, right?
So my thought about this is, if we are integrable, then having the p-th power of some real-value would not change being integrable, so infact these are equivalent. But since I'm a total novice in measure theory and I have made many mistakes where I was certainly sure,I'd like to see some approval.