\end{split}$$ where we have used $\mu_{\omega}$ to denote the probability measure of $\omega$ and write $${\mathbb{E}}_{\omega} f(\omega) = \int_{\Omega} f(\omega) \,\mathrm{d} \mu_{\omega}.$$ Note that the right hand side of involves the unknown wave function at time $t_1$. To proceed, we apply to its own right hand side and get $$\label{eq:int_form2}
\begin{split}
& \ket{\Psi(t)} = \mathrm{e}^{-{\mathrm{i}}t H_{\mathrm{easy}}} \ket{\Psi(0)} -
  {\mathrm{i}}\int_0^t \int_{\Omega} \mathrm{e}^{-{\mathrm{i}}(t-t_1) H_{\mathrm{easy}}}
    A(\omega_1) \mathrm{e}^{-{\mathrm{i}}t_1 H_{\mathrm{easy}}} \ket{\Psi(0)} 
  \,\mathrm{d}\mu_{\omega} \,\mathrm{d}t_1 \\
& + (-{\mathrm{i}})^2 \int_0^t \!\! \int_{\Omega} \int_0^{t_1} \!\!\! \int_{\Omega}
  \mathrm{e}^{-{\mathrm{i}}(t-t_1) H_{\mathrm{easy}}} A(\omega_1)
  \mathrm{e}^{-{\mathrm{i}}(t_1-t_2) H_{\mathrm{easy}}} A(\omega_2) \ket{\Psi(t_2)}
  \,\mathrm{d}\mu_{\omega_2} \,\mathrm{d}t_2 \,\mathrm{d}\mu_{\omega_1} \,\mathrm{d}t_1.
\end{split}$$ Inserting again into the right hand side of , we will get one more term on the right hand side. Such a substitution can be done repeatedly, which results into the following Dyson series expansion of $\ket{\Psi}$: $$\label{eq:Psi_expansion}
\begin{split}
  \ket{\Psi(t)} &= \sum_{M=0}^{+\infty} \int_0^t
  \int_{\Omega} \int_0^{t_1} \int_{\Omega} \cdots
  \int_0^{t_{M-1}} \int_{\Omega} (-{\mathrm{i}})^M \times \\
  & \times \mathrm{e}^{-{\mathrm{i}}(t-t_1) H_{\mathrm{easy}}}
  A(\omega_1) \mathrm{e}^{-{\mathrm{i}}(t_1-t_2) H_{\mathrm{easy}}}
  A(\omega_2) \cdots \\
  & \times \mathrm{e}^{-{\mathrm{i}}(t_{M-1}-t_M)
    H_{\mathrm{easy}}} A(\omega_M) \mathrm{e}^{-{\mathrm{i}}t_M
    H_{\mathrm{easy}}} \ket{\Psi(0)} \,\mathrm{d}\mu_{\omega_M}
  \,\mathrm{d}t_M \cdots
  \,\mathrm{d}\mu_{\omega_2} \,\mathrm{d}t_2 \,\mathrm{d}\mu_{\omega_1} \,\mathrm{d}t_1 \\
  &= \sum_{M=0}^{+\infty} \int_0^t \int_{\Omega} \int_0^{t_M}
  \int_{\Omega} \cdots
  \int_0^{t_2} \int_{\Omega} (-{\mathrm{i}})^M \times \\
  & \times \mathrm{e}^{-{\mathrm{i}}(t-t_M) H_{\mathrm{easy}}}
  A(\omega_M) \mathrm{e}^{-{\mathrm{i}}(t_M-t_{M-1}) H_{\mathrm{easy}}}
  A(\omega_{M-1}) \cdots \\
  & \times \mathrm{e}^{-{\mathrm{i}}(t_2-t_1) H_{\mathrm{easy}}}
  A(\omega_1) \mathrm{e}^{-{\mathrm{i}}t_1 H_{\mathrm{easy}}} \ket{\Psi(0)}
  \,\mathrm{d}\mu_{\omega_1} \,\mathrm{d}t_1 