Difference in reference of L space in Fubini Tonelli

I think my understanding of how $L^+$ and $L^1$ spaces are defined (I'm using Folland) is a little hazy. For example, in the Fubini-Tonelli theorem:

$\textbf{For the Tonelli part:}$ We start with if $f \in L^+(X \times Y)$ for which my understanding is the space of all measurable functions $f:X \times Y \rightarrow [0,\infty]$

$\textbf{For the Fubini part:}$ We start with if $f \in L^1(\mu \times \nu)$ which my understanding is the space of all complex valued integrable...product measures on $\mathcal{M} \otimes \mathcal{N}$?

Other than being unsure of what $f \in L^1(\mu \times \nu)$ means, my general question is why do we define $L^+$ spaces in terms of the domain set and $L^1$ spaces in terms of the measure? Or is this something just specific to the theorem (which I am then obviously missing)?

Many Thanks!

It makes sense that the notation of all measurable functions $f:X\times Y\to [0,\infty]$ does not involve the measure $\mu\times\nu$, because measurability only depends on which sigma-algebra you have put on $X\times Y$ (which usually is the product sigma-algebra $\mathcal M\otimes\mathcal N$). So one could maybe argue that $L^+(\mathcal M\otimes\mathcal N)$ was more natural, and maybe that something else than $L$ could have been used here.
As for $L^1(\mu\times\nu)$, it makes perfect sense that it involves the measure (and yes, this would be on the measure space $(X\times Y,\mathcal M\otimes\mathcal N,\mu\times\nu)$), because being an integrable function very much depend on the measure: the condition is that $$\int_{X\times Y}|f|\,\mathrm d(\mu\times\nu)<\infty.$$