Let $u$ be a subharmonic function in a domain $\Omega$ pf $\mathbb{C}$. The functions $u_{j} := \max(u, -j)$ still subharmonic. Let $\mu := \Delta u$ and $\mu_{j} := \Delta u_{j}$ be the associated Riesz measures (which are positive). Let $B$ be a borelian of $\Omega$.

**Question :** is it true that
$$
\int_{B} 1_{\{\phi > -j\}}\mu_{j} \to \int_{B} 1_{\{\phi > -\infty\}}\mu\quad\text{ as }\quad j \to +\infty\;?
$$

I try to prove it in the following way : let $B$ be a borelian. We then have

$$
\begin{split}
1_{\{\phi > -\infty\}}( \mu )(B) - 1_{\{\phi > -j\}}(\mu_{j})(B) & = 1_{\{\phi > -\infty\}}(\mu )(B) - 1_{\{\phi > -j\}}(\mu )(B) \\
&\quad + 1_{\{\phi > -j\}}(\mu)(B) - 1_{\{\phi > -j\}}(\mu_{j})(B)
\end{split}
$$
The quantity $1_{\{\phi > -\infty\}}(\mu)(B) - 1_{\{\phi > -j\} }(\mu)(B)$ converges to $0$ by monotone convergence. It then remains to estimate : $1_{\{\phi > -j\}}(\mu)(B) - 1_{\{\phi > -j\}}(\mu_{j})(B) = 1_{\{\phi > -j\}}(\mu - \mu_{j})(B)$. But I don't have enough information on the measures $\mu_{j}$ and $\mu$ to estimate this.

I wish you a good day.