I am reading a proof in Ransford's Potential Theory in the Complex Plane, where he uses the monotone convergence theorem on a negative function, and I do not understand why he can do that.
It is stated that for $m \geq 1$ and a compact subset $K$ of $\mathbb{C}$:
\begin{eqnarray*} \int_K \int_K \max(\log|z-w|,-m)\,d \mu (z) d \mu (w) \to \int_K \int_K \log|z-w|\,d \mu (z) d \mu (w), \end{eqnarray*} for $m \to \infty$, since $\max(\log|z-w|,-m) \to \log|z-w|$ for $m \to \infty$.
Can someone help me to see why this is possible?
The sequence of functions, $\min(-\log|z-w|,m)$ increases in $m$. Monotone convergence applies to them.
The integrals in the question are the negatives of the integrals of these functions.
Considerations from Comments
We are looking for $$ \begin{align} &\lim_{m\to\infty}\int_K\int_K\max\left(\log|z-w|,-m\right)\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)\\ &=-\lim_{m\to\infty}\int_K\int_K\min\left(-\log|z-w|,m\right)\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)\\ &=-\lim_{m\to\infty}\int_K\int_K[|z-w|\gt1]\min\left(-\log|z-w|,m\right)\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)\\ &\phantom{\,=\,}-\lim_{m\to\infty}\int_K\int_K[|z-w|\le1]\min\left(-\log|z-w|,m\right)\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)\\ &=\color{#090}{\int_K\int_K[|z-w|\gt1]\log|z-w|\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)}\\ &\phantom{\,=\,}-\color{#C00}{\lim_{m\to\infty}\int_K\int_K[|z-w|\le1]\min\left(-\log|z-w|,m\right)\,\mathrm{d}\mu(z)\,\mathrm{d}\mu(w)}\\ \end{align} $$ The green integral must converge if the integrals converge for any $m$. In fact, the green integral is the $m=0$ integral in the sequence. The red integral is a monotonically increasing sequence of non-negative functions; thus, we can apply the Monotone Convergence Theorem.