Let \(F_ r\) denote the free group with \(r\) generators. The author specifies a natural concept of irreducibility of a function \(f\) on \(F_ r\) under right translation. On the other hand, if \(\Omega\) is the ``boundary'' of \(F_ r\), there is the set \(M(\Omega)\) of all probability measures on \(\Omega\) and the author proves that \(f\) is irreducible and continuous if and only if there exists \(m\in M(\Omega)\) such that \[ f(x)=\int_ \Omega f(x\omega)dm(\omega). \] A function \(f\) is said to be \(\mu\)-harmonic for a measure \(\mu\in M(\Omega)\) if \(f=f*\mu\) (convolution). If the convolution powers \(\mu^ n(x)\to 0\), \(\forall x\in F_ r\), then a \(\mu\)-harmonic function is irreducible, but not conversely; there are two interesting counterexamples.