  We obtain new integral inequalities for the integrals of the difference of
subharmonic functions in measure through their Nevanlinna characteristic and
some functional characteristic of the measure. These results are new also for
meromorphic functions. Let us illustrate the main result for the case of a
meromorphic function. We denote by $D_z(r)$ a closed disc of radius $r$
centered at $z$ in the complex plane $\mathbb C$. Let $\mu\geq 0$ be a Borel
measure on $D_0(r)$, and $M:=\mu(D_0(r))<+\infty$. The modulus of continuity of
measure $\mu$ is the function $h(t):=\sup_{z\in \mathbb C}\mu(D_z(t)), \quad
t\geq 0.$ Suppose that $\int_0\frac{h(t)}{t} dt<+\infty$. Let $f\neq 0$ be a
meromorphic function on a neighbourhood of $D_0(R)$ of radius $R>r$ with
$f(0)\in \mathbb C$ and the Nevanlinna characteristic $T_f$ . Then there is
$$\int \ln^+|f| \,d\mu \leq
2\frac{R+r}{R-r}T_f(R)M\biggl(1+\int_0^{R+r}\frac{h(t)}{t} dt\biggr).$$
