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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:

  1. 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 $L^p$.

  2. 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?

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

  4. 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 $\| S(P,x)-I\|_p<\epsilon$.

  5. 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 $\| S(P,x)-I\|_p<\epsilon$.

  6. 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...

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I have a partial answer to my questions. Let us consider the following general definition, which I will refer to as DEF: $x$ is $L^p$-Riemann integrable on $[a,b]$ if there exists 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 that $\lim_{n\rightarrow\infty} \sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)$ exists in $L^p$.

Under this definition DEF, it is easy to prove that if each Riemann sum $\sum_{i=1}^{r_n} x(s_i^n)(t_i^n-t_{i-1}^n)$ converges in $L^p$, then all the limits must be a common random variable $A$. This $A$ is denoted as $\int_a^b x(t)\,dt$. Indeed, if for two sequences of interior points $\{s_{i1}^n\}$ and $\{s_{i2}^n\}$ we had two different limits for the Riemann sums, say $A_1$ and $A_2$, then the Riemann sum constructed with interior points $\{s_i^n\}$ defined as $s_i^n=s_{i1}^n$ if $n$ odd, $s_i^n=s_{i2}^n$ if $n$ even, would have two subsequences that converge to $A_1$ and $A_2$, so necessarily $A_1=A_2$ a.s. to preserve convergence (see https://math.stackexchange.com/questions/3182682/question-about-riemann-integrability-do-we-need-to-specify-that-all-riemann-sum for the original idea). This reasoning justifies 2.

Now, with definition DEF, one can prove that $x$ is $L^p$-Riemann integrable on $[a,b]$ if and only if $\Gamma_X(t_1,\ldots,t_p):=E[x(t_1)\cdots x(t_p)]$ is multidimensional Riemann integrable on $[a,b]^p$. Hence, Theorem 4.5.1 in Soong works (although the proof therein is informal$\ldots$). This is based on the fact that a sequence of random variables $\{y_n\}_{n=1}^\infty$ converges in $L^p$ if and only if $E[y_{n_1}\cdots y_{n_p}]$ converges as $n_1,\ldots,n_p\rightarrow\infty$.

Notice that DEF implies that $\Gamma_X$ is Riemann integrable on $[a,b]^p$, which implies that $\Gamma_X$ is bounded on $[a,b]^p$. This is equivalent to $\|x(t)\|_p=(\Gamma_X(t,\ldots,t))^{1/p}$ being bounded on $[a,b]$ (by Hölder's inequality). That is, DEF implies that $x$ is $L^p$-bounded on $[a,b]$.

Notice that we obtain 1 for free, as we know that for any real Riemann integrable function, such as $\Gamma_X(t_1,\ldots,t_p)$, we can take any sequence of partitions.

For 3, I do not have a complete answer. I think that the answer is no, as we do not have the concept of supremum and infimum in this random setting$\ldots$

Question 6 is very interesting, as it is the extension of the Lebesgue's criterion for Riemann integrability. A similar discussion appeared in https://math.stackexchange.com/questions/2465113/showing-that-the-norm-of-a-riemann-integrable-banach-valued-function-is-a-real-v and https://math.stackexchange.com/questions/163367/lebesgues-criterion-for-riemann-integrability-of-banach-space-valued-functions:

  • If $x$ is $L^p$-bounded and almost everywhere $L^p$-continuous on $[a,b]$, then $x$ is $L^p$-Riemann integrable: Indeed, we saw above that $\Gamma_X$ is bounded on $[a,b]$; and on the other hand, there is a null set $N\subseteq[a,b]$ such that, for all $t\notin N$, $x(t+\tau)\rightarrow x(t)$ as $\tau\rightarrow0$, therefore $\Gamma_X$ is continuous on $([a,b]\backslash N)^p$, that is, $\Gamma_X$ is almost everywhere continuous on $[a,b]^p$, which implies its Riemann integrability on $[a,b]^p$ and the $L^p$-Riemann integrability of $x$ on $[a,b]$.

  • However, the other implication is not true. If $x$ is $L^p$-Riemann integrable, then $x$ is certainly $L^p$-bounded on $[a,b]$. But $L^p$-Riemann integrability does not imply almost everywhere $L^p$-continuity. See https://warwick.ac.uk/fac/sci/maths/research/events/2014-15/nonsymposium/rbst/prog/martinezcervantes.pdf or https://arxiv.org/pdf/1510.08801.pdf, for instance. A Banach space $X$ is said to have the Lebesgue Property if every Riemann integrable function $f:[a,b]\rightarrow X$ is almost everywhere continuous. The Lebesgue space $L^p$, for $1\leq p<\infty$, does not have the Lebesgue Property.

Statements 4 and 5 are equivalent. See Definition 2 and Theorem 3 in (Gordon R., Survey Article Riemann Integration in Banach Spaces, Rocky Mountain Journal of Mathematics, 21(3), 923–949 (1991), doi:10.1216/rmjm/1181072923).

Clearly, 4 (and 5) implies DEF. It would remain to prove DEF $\Rightarrow$ 4. See https://math.stackexchange.com/questions/3184137/two-definitions-for-riemann-integrability-of-fa-b-rightarrow-x-where-x-i/3185220#3185220 for the original idea. By Theorem 5 condition (4) in (Gordon R., Survey Article Riemann Integration in Banach Spaces, Rocky Mountain Journal of Mathematics, 21(3), 923–949 (1991), doi:10.1216/rmjm/1181072923), statement 4 is equivalent to: for each $\epsilon>0$, there exists a partition $P_\epsilon$ of $[a,b]$ such that $\|S(P_1,x)-S(P_2,x)\|_p<\epsilon$ for all tagged partitions $P_1$ and $P_2$ of $[a,b]$ with the same points as $P_\epsilon$. Assume DEF and suppose that this condition is not satisfied. There exists $\epsilon_0>0$ such that, for every partition $P$ of $[a,b]$, there exist two tagged partitions $P_1$ and $P_2$ of $P$ with $\|S(P_1,x)-S(P_2,x)\|_p\geq\epsilon_0$. In particular, for each $n\geq1$, we can choose two taggs in $P_n$, say $P_n^1$ and $P_n^2$, such that $\|S(P_n^1,x)-S(P_n^2,x)\|_p\geq\epsilon_0$. But $\lim_{n\rightarrow\infty}S(P_n^1,x)=I=\lim_{n\rightarrow\infty}S(P_n^2,x)$ in $L^p$, which is therefore a contradiction.

In several variables, analogous results hold. For example, $x(t,s)$ is $L^p$-Riemann integrable on $[a,b]\times[c,d]$ if and only if $E[x(t_1,s_1)\cdots x(t_p,s_p)]$ is real Riemann integrable on $[a,b]\times[c,d]\times\cdots \times[a,b]\times[c,d]$.

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