It was part of our definition 1.1 that $\omega$-psh functions are integrable with respect to a fixed volume form. Therefore $PSH(X,\omega)\subset L^{1}(\mu)$ for every smooth probability measure $\mu$ on $X$. More generally if $\mu$ is a probability measure on $X$ such that

(1)

$$\mu=\Theta+dd^{c}(S),$$

where $\Theta$ is smooth and $S$ is a positive current of bidimension $(1,1)$ on $X$, then $PSH(X,\omega)\subset L^{1}(\mu)$ for any smooth $\omega$. Indeed let $\varphi$ in $PSH(X,\omega)$, $\varphi\leq 0$. If $\varphi$ is smooth, it follows from Stokes theorem that

	$\displaystyle 0\leq\int_{X}(-\varphi)d\mu$	$\displaystyle=$	$\displaystyle\int_{X}(-\varphi)\Theta+\int_{X}(-\varphi)dd^{c}S$
		$\displaystyle\leq$	$\displaystyle C_{\Theta}||\varphi||_{L^{1}}+\int_{X}S\wedge(-dd^{c}\varphi)$
		$\displaystyle\leq$	$\displaystyle C_{\Theta}||\varphi||_{L^{1}}+\int_{X}S\wedge\omega<+\infty,$

where the last inequality follows from $S\geq 0$ and $-dd^{c}\varphi\leq\omega$. The general case follows by regularizing $\varphi$ (see Appendix).

Probability measures satisfying $(1)$ naturally arise in complex dynamics (see [23]). Observe also that Monge-Ampère measures arising from the local theory of Bedford and Taylor [5] do satisfy $(1)$: if $u$ is psh and locally bounded near e.g. the unit ball $B$ of $\mathbb{C}^{n}$, we can extend it to $\mathbb{C}^{n}$ as a global psh function with logarithmic growth considering

$$U(z):=\left\{\begin{array}[]{rl}u(z)&\text{ if }z\in B\\ \max(u(z),A\log^{+}|z|-\sup_{B}|u|-1)&\text{ if }z\in(1+\varepsilon)B\setminus B\\ A\log^{+}|z|-\sup_{B}|u|-1&\text{ if }z\in\mathbb{C}^{n}\setminus(1+\varepsilon)B\end{array}\right.$$

where $\log^{+}|z|:=\max(\log|z|,0)$ and with $A$ large enough. We assume $A=1$ for simplicity. Now $\varphi:=U-\frac{1}{2}\log[1+|z|^{2}]+C$ extends as a bounded function in $PSH(\mathbb{C}\mathbb{P}^{n},\omega)$, where $\omega$ is the Fubini-Study Kähler form on $\mathbb{C}\mathbb{P}^{n}$, so $\varphi\geq 0$ if $C>0$ is large enough. To conclude note that, setting $\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$, we get $\omega_{\varphi}^{n}=(dd^{c}u)^{n}$ in $B$ and

$$\omega_{\varphi}^{n}=\omega^{n}+dd^{c}S,\text{ where }S=\varphi\sum_{j=0}^{n-1}\omega_{\varphi}^{j}\wedge\omega^{n-1-j}\geq 0.$$

The Monge-Ampère operator $\omega_{\varphi}^{n}$ will be defined in the next section.