A remark on transition probability

Uhlmann's transition probability P(ψ, φ) of two normal states of a von Neumann algebra M, which is the supremum of |(Ψ, Φ)|² for all possible choices of representative vectors Ψ and Φ of ψ and φ, is shown to be the infimum of (∫d(µ_{ψ, e})^1/2)² for the induced measures µ_{ω, e}(B)=ω(e(B)) (B: Borel set in ?, ω=ψ, φ) for all possible projection-valued measures e belonging to M.