 Suppose ("X", "M", μ) is a measure space and let {"H""x"}"x" ∈ "X"  be a μ-measurable family of separable Hilbert spaces.For every "E" ∈ "M", let π("E") be the operator of multiplication by 1"E" on the Hilbert space

Then π is a projection-valued measure on ("X", "M").Suppose π, ρ are projection-valued measures on ("X", "M") with values in the projections of "H", "K".