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<math display="block"> \mu_h = \mu_{\mathrm{ac}} + \mu_{\mathrm{sc}} + \mu_{\mathrm{pp}}</math>
where ''μ''<sub>ac</sub> is absolutely continuous with respect to the Lebesgue measure, ''μ''<sub>sc</sub> is singular with respect to the Lebesgue measure and atomless, and ''μ''<sub>pp</sub> is a pure point measure.{{sfn|Simon|2005|page=43}}
All three types of measures are invariant under linear operations. Let ''H''<sub>ac</sub> be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the [[Lebesgue measure]]. Define ''H''<sub>pp</sub> and ''H''<sub>sc</sub> in analogous fashion. These subspaces are invariant under ''T''. For example, if ''h'' ∈ ''H''<sub>ac</sub> and ''k'' = ''T h''. Let ''χ'' be the characteristic function of some Borel set in ''σ''(''T''), then
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