I have been studied a research for many days, but I have been confused by mathematical inequalities. I list the calculations as below and explain the notations.
\begin{equation} \begin{split} \left|{\hat{Q}}_N\left(\gamma,F_1\right)-{\hat{Q}}_N\left(\gamma,F_2\right)\right|\le2\frac{\sum_{i=1}^{N}\sum_{j=1}^{J}\sum_{t=1}^{T}\pi_{ijt}\left|\int{g_j\left(x_i,\beta,\gamma\right)\left(dF_1\left(\beta\right)-dF_2\left(\beta\right)\right)}\right|}{NJT} +\frac{1}{NJT}\sum_{i=1}^{N}\sum_{j=1}^{J}\sum_{t=1}^{T}\int{g_j\left(x_i,\beta,\gamma\right)\left(dF_1\left(\beta\right)+dF_2\left(\beta\right)\right)}\\ \times|\int{g_j\left(x_i,\beta,\gamma\right)\left(dF_1\left(\beta\right)-dF_2\left(\beta\right)\right)|} \\ \le4\sum_{i=1}^{N}\sum_{j=1}^{J}\sum_{t=1}^{T}\left|\int{g_j\left(x_i,\beta,\gamma\right)\left(dF_1\left(\beta\right)-dF_2\left(\beta\right)\right)}\right|/NJT \end{split} \end{equation}
${\hat{Q}}_N$ is the criteria function. \begin{equation} {\hat{Q}}_N\left(\gamma,F_1\right)=\frac{1}{NJ}\sum_{i=1}^N\left\|y_i-\int g(x_i,β)dF(β))\right\|_E^2=\frac{1}{NJ}\sum_{i=1}^N\left\|y_i-\sum_{r=1}^Rθ^rg(x_i,β^r)\right\|_E^2 \end{equation} \begin{equation} \end{equation}
$\pi_{ijt} = [0,1]$
$\gamma$ is the distribution parameter for $F$
$g(x_i,\beta,\gamma)$ is a choice probability
Thnaks for reading this very long equations. I understand the second inequality holds due to that $y_i,g_j(x_i,\beta,\gamma) and \int g_j(x_i,\beta,\gamma) dF(\beta)$ are uniform bounded by 1.
My questions are below：
1.Why two integral of cumulative function $\int g(\cdot) dF_1-\int g(\cdot) dF_2$ can be combine into $\int g(\cdot) (d(F_1-F_2))$
2.According to the paper, the first inequality holds by using of triangle inequality. I can't link the triangle inequality with this form. Is it related to $\int g(\cdot) (d(F_1-F_2))$ and $\int g(\cdot) (d(F_1+F_2))$ ?
Hope anyone can help me understand the mathematical calculation. Thanks.