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I came across the following identity $$ \lim_{\Delta x\Delta t\to 0}\prod_{x,t\in A}(1-f(x,t) \Delta x\Delta t)=\exp\left(-\int\int_{x,t\in A}f(x,t)\, dxdt\right) $$ where $A$ is some subset of $\mathbb{R}^2$. Is this true?

Thoughts: This seems to be a type I Volterra product integral. On the linked page, we have $$ \prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right) $$ In our case, $\mu$ would be a Lebesgue measure and we could simply redefine $f(x,t)\equiv -f(x,t)$. However, is the Lebesgue measure here well defined? Perhaps it's just a matter of notation ($\Delta x, \Delta y$ can be interpreted the same as $dxdt$), or am I missing some important detail?

Mittens
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sam wolfe
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  • Quick answer: $\iint_{x,t\in A}f(x,t),dx,dt=\lim\limits_{\Delta x\Delta t\to 0}\sum\limits_{x,t\in A}f(x,t),\Delta x,\Delta t,.$ Now use $e^{a+b}=e^ae^b$ and $e^{-c}\approx 1-c$ for small $c,.$ – Kurt G. Feb 01 '24 at 17:33

1 Answers1

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I tried to make sense of the product integrals as given by the OP and the link he provided. I can prove the statement for functions $f$ that are Riemann integrable in some box $C=[a,b]\times[c,d]\subset\mathbb{R}^2$ as follows.

Let $\mathcal{P}:=\{[t_i,t_{i+1}]\times[s_j,s_{j+1}]: 0\leq I< n, 0\leq j< m\}$ be a partition of $[a,b]\times [c,d]$ ($a=t_0<\ldots<t_n=b$, $c=s_0<\ldots<s_m=d\}$. Assume $|f(\mathbf{x})|\leq M$, for all $\mathbf{x}\in C$. Then for any $A\in\mathcal{P}$, choose a tag $\mathbf{x}_A\in A$ and define $$Q_{\mathcal{P}}=\prod_{A\in\mathcal{P}}\big(1+f(\mathbf{x}_A)\lambda_2(A)\big)$$ where $\lambda_2([t_i,t_{i+1}]\times[s_j,s_{j+1}])=(t_{i+1}-t_i)(s_{j+1}-s_j)$ if $A=[t_i,t_{i+1}]\times[s_j,s_{j+1}]$.

Suppose partitions $\mathcal{P}_k$ are taken so that $\max_{A\in\mathcal{P}_k}\lambda_2(A)=0$.

Then

  • $\max_{A\in\mathcal{P}_k}|f(\mathbf{x}_A)\lambda_2(A)|\xrightarrow{k\rightarrow\infty}0$

  • $\lim_k\sum_{A\in\mathcal{P}_k}f(\mathbf{x}_A)\lambda_2(A)=\int_Cf$

  • $\sup_k\sum_{A\in\mathcal{P}_k}|f(\mathbf{x}_A)|\lambda_2(A)\leq M\lambda_2(C)$

Then, $\lim_kQ_{P_k}=\exp\Big(\int_Cf\Big)$. This follows from Theorem A below. The result can be extended to Riemann integrable functions over bounded closed boxes on $\mathbb{R}^d$. For more general functions, for example, Lebesgue integrable the statement in the OP may not hold.

Theorem A: Let $\{c_{n,m}:1\leq m\leq m_n\}\subset\mathbb{C}$. Suppose that

  1. $\lim\limits_{n\rightarrow0}\sup_{1\leq m\leq m_n}|c_{n,m}|=0$,
  2. $\lim\limits_{n\rightarrow\infty}\sum^{m_n}_{m=1}c_{n,m}=c\in\mathbb{C}$,
  3. and $M:=\sup_n\sum^{m_n}_{m=1}|c_{n,m}|<\infty$. Then \begin{align}\prod^{m_n}_{m=1}(1+c_{n,m})=e^c \end{align}

A proof pot this result can be found here

Mittens
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