# Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$

Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} \int_{\Delta_{a,x_i}^m} f(x_1,\dots, x_n) g(y_1,\dots, y_m) dy_m \cdots dy_1 dx_n \cdots dx_1$$ for some fixed $i\in \{1,\dots,n\}$ and here $f$ and $g$ are integrable functions. I have the feeling that one can rewrite this double integral above as:

$$I= \sum_{j=1}^i \int_{a< x_1 < \cdots < x_{j-1} < y_1<\cdots <y_m<x_j < \cdots x_n < b} f(x_1,\dots, x_n) g(y_1,\dots, y_m) dx_n\cdots dx_j dy_m \cdots dy_1 dx_{j-1}\cdots dx_1$$ where we set $x_0:=a$, i.e. we divide up the inner integral in sums. Is this correct? Because, on the other hand if $x_i=b$ in the integral $I$ we have \begin{align*} I=&\left(\int_{\Delta_{a,b}^n} f(x_1,\dots, x_n)dx_n\cdots dx_1\right)\left(\int_{\Delta_{a,b}^m} g(y_1,\dots, y_m)dy_m\cdots dy_1\right)\\ =& \sum_{\sigma \in shuffle(n,m)}\int_{\Delta_{a,b}^{n+m}} f(w_{\sigma(1)},\dots, w_{\sigma(n)})g(w_{\sigma(n+1)},\dots, w_{\sigma(n+m)}) dw_{n+m}\cdots dw_1 \end{align*} where $Shuffle(n,m)$ is the set of all permutations of $\{1,\dots, n+m\}$ such that $\sigma(1)<\cdots <\sigma(n)$ and $\sigma(n+1)<\cdots <\sigma(n+m)$.

Maybe both are correct?

The division of the integral in sums is incorrect, because of the domain of the integration: not necessarily all $y$’s are between consecutive $x$’s.
Also there are misprints in upper indices of $\Delta$ in the integral formulas with “shuffle”. (And I did not check, is this formula correct or not).
• You are right, it said $\Delta_{a,b}^m$ but of course it should say $\Delta_{a,b}^{n+m}$ since the integral is then doubled. Then you agree that the overall first integral can be written as a sum of an integral over a bigger oredered domain by shuffling right? How would it look like? I have a guess but I preferred not to include it. – Martingalo Jan 25 '15 at 9:51
• @Martingalo Maybe you are right and this sum corresponds to a partition of the integration domain $\Delta^n_{a,b}\times \Delta^m_{a,b}$, but I don’t see this sufficiently clear. – Alex Ravsky Jan 25 '15 at 17:11