In functional analysis, let $Dom A$ be Hilbert space $\mathcal{H}$. Let A be self-adjoint operator, $A = A^*$, $m_A > 0$. Here $m_A = \inf_{u \in Dom A} \frac{(Au, u)}{|u|^2_\mathcal{H}}$. $E$ is spectral measure which maps Borel sets of real line to set of bounded operators, i.e. $E: \mathcal{B} \rightarrow B(\mathcal{H})$.
There exists such a formula $A = \int_{\mathcal{R}} \lambda \, dE_A(\lambda) = \int_{m_A}^{\infty} \lambda \, dE_A(\lambda)$. How to prove this?