In Kadison and Ringrose's book "FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS", the author gives the following theorem.
Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space $\mathscr{H}$ and $\mathscr{A}$ is an abelian von Neumann algebra containing $A$, there is a family $\{E_\lambda\}$ of projections, indexed by $\mathbb{R}$, in $\scr{A}$ such that
(1) $E_\lambda=0$ if $\lambda<-\|A\|$ and $E_\lambda=I$ if $\lambda\geq\|A\|$;
(2) $E_\lambda\leq E_{\lambda'}$ if $\lambda\leq\lambda'$;
(3) $E_\lambda=\wedge_{\lambda'>\lambda}E_{\lambda'}$;
(4) $AE_\lambda\leq\lambda E_\lambda$ and $A(I-E_\lambda)\leq\lambda(I-E_\lambda)$;
(5) $A=\int^{\|A\|}_{-\|A\|}\lambda dE_\lambda$ in the sense of norm convergence of approximating Riemann sums; and $A$ is the norm limit of finite linear combinations with coefficients in $sp(A)$ of orthogonal projections $E_{\lambda'}-E_\lambda$.
I want to know what (3) means and also what is the relationship between such so called "spectral resolution" $E_\lambda$ and the spectral measure?
The formula connecting the spectral resolution of $E_{\lambda}$ and the spectral measure $E$ is $E_\lambda=E((-\infty,\lambda]),\ \mbox{for all}\ \lambda\in\mathbb R.$ You can get $E$ from $E_\lambda$ by setting $E((a,b]):=E_b-E_a$ etc.
Condition (3) means $E_\lambda$ is the greatest lower bound of $E_{\lambda'},\ \lambda'>\lambda.$ Equivalently, $E_\lambda$ is the strong limit of $E_{\lambda'},\ \lambda'\to\lambda+0.$