Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $\Omega$ with finite absolute moment of order $p$, with norm $\|\cdot\|_p$.

Consider the following definition of Riemann integrability in the sense of $L^p$: we say that $x$ is $L^p$-Riemann integrable on $[a,b]$ if there is a random variable $I$ and a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, such that, for any choice of interior points $s_i^n\in [t_{i-1}^n,t_i^n]$, we have $\lim_{n\rightarrow\infty} \sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)=I$ in $L^p$. In this case, $I$ is denoted as $\int_a^b x(t)\,dt$. This approach is defined in (T.T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973) and (T.L. Saaty, Modern Nonlinear Equations, Dover Publications Inc., New York, 1981), for instance.

I have not been able to read a full exposition on $L^p$-Riemann integration anywhere. I have several questions regarding this definition:

Once we know that $x$ is $L^p$-Riemann integrable and that such a especial sequence of partitions $\{P_n\}_{n=1}^\infty$ exists, can we take any other sequence of partitions $\{P_n'\}_{n=1}^\infty$? I mean, for any $\{P_n'\}_{n=1}^\infty$ with mesh tending to $0$ and any choice of interior points, the corresponding Riemann sums tends to $I$ in $L^p$.

If I merely know that $\lim_{n\rightarrow\infty} \sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)$ exists, for any choice of interior points $s_i^n\in [t_{i-1}^n,t_i^n]$, can I prove that those limits coincide?

Can this definition be related to upper and lower sums, as one does in real integration with the Darboux integral?

Equivalence with this statement: there is a random variable $I$ such that: for every $\epsilon>0$, there is a partition $P_\epsilon$ such that, for every partition $P$ finer than $P_\epsilon$ and for any choice of interior points, the corresponding Riemann sum $S(P,x)$ satisfies $\| S(P,x)-I\|_p<\epsilon$.

Equivalence with this statement: there is a random variable $I$ such that: for every $\epsilon>0$, there is a $\delta>0$ such that for any partition $P$ with $\|P\|<\delta$ and for any choice of interior points, the corresponding Riemann sum $S(P,x)$ satisfies $\| S(P,x)-I\|_p<\epsilon$.

Equivalence with: $x$ is $L^p$-bounded on $[a,b]$ and almost everywhere $L^p$-continuous on $[a,b]$.

And now we move to several variables. Let $x:[a,b]\times[c,d]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process. Briefly, we say that $x$ is $L^p$-Riemann integrable on $[a,b]\times[c,d]$ if there is a random variable $I$ and a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$ such that, for any choice of interior points, the corresponding Riemann sums tend to $I$ in $L^p$. In such a case, $I$ is denoted $\iint_{[a,b]\times[c,d]} x(t,s)\,dt\,ds$. Do the above questions hold for this double integral as well? And another question in this particular setting: consider two partitions $\{P_n'\}_{n=1}^\infty$ and $\{P_m''\}_{m=1}^\infty$ of $[a,b]$ and $[c,d]$, respectively, with mesh tending to $0$, and let $P_{n,m}=P_n'\times P_m''$. Do the Riemann sums corresponding to $P_{n,m}$ converge to $I$ in $L^p$ as $n,m\rightarrow\infty$ (in the sense of double sequences)?

In case of someone using the characterization from Theorem 4.5.1 in Soong: $x$ is $L^p$-Riemann integrable if and only if $E[x(t_1)\cdots x(t_p)]$ is real Riemann integrable on $[a,b]^p$. I am not sure if this statement from the book is correct... For instance, assume $x$ is $L^p$-Riemann integrable. There is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, such that, for any choice of interior points $s_i^n\in [t_{i-1}^n,t_i^n]$, we have $\lim_{n\rightarrow\infty} \sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)=I$ in $L^p$. Let $y_n=\sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)\rightarrow I$. This implies $E[y_{n_1}\cdots y_{n_p}]\rightarrow E[I^p]$ as $n_1,\ldots,n_p\rightarrow\infty$. Now, $E[y_{n_1}\cdots y_{n_p}]$ is a Riemann sum for $E[x(t_1)\cdots x(t_p)]$, but its interior points are not arbitrary for $(P_n)^p=P_n\times \cdots \times P_n$. So one cannot guarantee that $E[y_{n_1}\cdots y_{n_p}]$ is Riemann integrable, I think...