Proof of Lemma 3.4 . [01A5]

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Proof of Lemma 3.4.

Note first that $\psi$ may be assumed to be a model function by

Lemma 3.5.

Let $\nu$ be a positive Radon measure on $X$ and let $\varphi$ be a bounded $\theta$ -psh function. Then we have

\int\varphi\nu=\inf_{\psi\geq\varphi}\int\psi\nu

where $\psi$ ranges over all $\theta$ -psh model functions such that $\psi\geq\varphi$ .

Since we already know that $(\varphi_{1},\dots,\varphi_{p})\mapsto\MAC(\varphi_{1},\dots,\varphi_{p})$ is continuous along decreasing nets, we may by regularization assume that all $\varphi_{i}$ and $\chi_{i}$ are also model functions. Integration by parts (3.1) then yields

\int\psi\MAC(\chi_{1},\chi_{2},\dots,\chi_{p})-\int\psi\MAC(\varphi_{1},\chi_{2},\dots,\chi_{p})=\\ =\int(\chi_{1}-\varphi_{1})\MAC(\psi,\chi_{2},\dots,\chi_{p})-\int(\chi_{1}-\varphi_{1})\MAC(0,\chi_{2},\dots,\chi_{p})

hence

\int\psi\MAC(\chi_{1},\dots,\chi_{p})\geq\int\psi\MAC(\varphi_{1},\chi_{2},\dots,\chi_{p})+\int(\varphi_{1}-\chi_{1})\MAC(0,\chi_{2},\dots,\chi_{p}).

We similarly have

	$\displaystyle\int\psi\MAC(\varphi_{1},\chi_{2},\chi_{3},\dots,\chi_{p})$	$\displaystyle\geq\int\psi\MAC(\varphi_{1},\varphi_{2},\chi_{3},\dots,\chi_{p})$
		$\displaystyle+\int(\varphi_{2}-\chi_{2})\MAC(\varphi_{1},0,\chi_{3},\dots,\chi_{p}).$

Iterating this argument and summing up then yields the desired result. ∎