The following is a note taking from [SS00], Lemma 3.1 and [HTW06], Lemma 8.1. Consider the set of all dyadic subintervals of $[0, 1]$ . If $I$ and $J$ are two such subintervals, we define the relation $I \sim J$ if $I$ and $J$ have the same length, $I$ is to the left of $J$, and that $I$ and $J$ are not adjacent but have adjacent parents (i.e., $I \subset I_0$ and $J \subset J_0$ where $I_0$ and $J_0$ are adjacent dyadic intervals of twice the length). Note that if $J$ is ﬁxed there are at most two intervals with $I \sim J$. **Claim** For almost every $(x, y) \in [0, 1]^2$ with $x < y$ there is a unique pair $I, J$ with $I \sim J$ and $x \in I$ and $y \in J$. Now suppose $F(s): [a,b]\mapsto [0,1]$ is a bijection. Let $x = F(s)$ and $y = F(t)$ then we conclude that the following identity holds almost everywhere: $$\begin{split} \chi_{\{ (s,t): s<t\}} (s,t) &= \chi_{\{(x,y): x<y\}}(x,y) ~\\ &=\sum_{I,J; I\sim J}\chi_I(x)\chi_J(y) ~\\ &=\sum_{I,J; I\sim J} \chi_{ F^{-1}(I)}(s) \chi_{F^{-1}(J)}(t), \end{split}$$ where the second inequality essentially used the **Claim**. **Remark**We can also define the relation $I\sim J$ If $I$ and $J$ have the same length, $I$ is to the left of $J$, and that $I$ and $J$ are adjacent. Then the **Claim** also holds in this case. This relation is defined in [Tao00], Lemma 3.1. === References [SS00] H. Smith, C.D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. PDE **25** (2000), 2171–2183 [HTW06] A. Hassel, T. Tao, J. Wunsch, Sharp Strichartz estimates on non-trapping asymptotically conic manifolds, Amer. J. Math. **128** (2006) 963–1024 [Tao00] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrdinger equation, Comm. Partial Diﬀerential Equations **25** (2000), no. 7-8, 1471–1485.