$\int_{0}^{t}\sum_{i=1}^{n}dN_i(u)=?$

Question

Suppose $T,C\in\mathbb{R}_+.$

Let $U_i=\min(T_i,C_i), i=1,2,\ldots,n.$

Also let $\Delta_i=I(T_i

Define $N_i(u)=I(U_i\le u, \Delta_i=1).$

Then $$\int_{0}^{t}\sum_{i=1}^{n}dN_i(u)=?$$