Integral Representations and Decompositions of Operator Monotone Functions on the Nonnegative Reals

In this paper, we show that there is a one-to-one correspondence between operator monotone functions on the nonnegative reals and finite Borel measures on the unit interval. This correspondence appears as an integral representation of special operator monotone functions $x\mapsto 1{!}_{t}x$ for $t\in [0,1]$ with respect to a finite Borel measure on $[0,1]$, here ${!}_t$ denotes the $t$-weighted harmonic mean. Hence such functions form building blocks for arbitrary operator monotone functions on the nonnegative reals. Moreover, we use this integral representation to decompose operator monotone functions.