Let $(X,\mathcal{F})$ be a measurable space and let $E:\mathcal{F}\to\mathscr{B(H)}$ be a spectral measure. Let $\phi\in B(X)$ be a simple function whose image is $\{\lambda_1,\ldots,\lambda_n\}\subset\mathbb{C}$, define
$\intop_X \phi dE = \sum_{i=1}^n\lambda_i\cdot E(\phi^{-1}(\lambda_i))$
Now, for a general $\phi\in B(X)$ let $\phi_n\to\phi$ uniformly, and define
$\intop_X \phi dE = \lim\intop_X \phi_n dE$
Prove that this integration operator is uniquely defined.