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Proof.
As in [GZ07 , Theorem 1.5] the result easily follows from the locality property by integration. More precisely, for any ε > 0 \varepsilon>0
1 = ∫ MA ( max { φ , ψ − ε } ) ≥ ∫ { φ < ψ − ε } MA ( max { φ , ψ − ε } ) + ∫ { φ > ψ − ε } MA ( max { φ , ψ − ε } ) = ( 5.1 ) ∫ { φ < ψ − ε } MA ( ψ − ε ) + ∫ { φ > ψ − ε } MA ( φ ) = ∫ { φ < ψ − ε } MA ( ψ ) + 1 − ∫ { φ ≤ ψ − ε } MA ( φ ) , 1=\int\MA(\max\{\varphi,\psi-\varepsilon\})\geq\int\limits_{\{\varphi<\psi-\varepsilon\}}\MA(\max\{\varphi,\psi-\varepsilon\})+\int\limits_{\{\varphi>\psi-\varepsilon\}}\MA(\max\{\varphi,\psi-\varepsilon\})\\
\mathop{=}\limits^{\eqref{eq:compar}}\int\limits_{\{\varphi<\psi-\varepsilon\}}\MA(\psi-\varepsilon)+\int\limits_{\{\varphi>\psi-\varepsilon\}}\MA(\varphi)=\int\limits_{\{\varphi<\psi-\varepsilon\}}\MA(\psi)+1-\int\limits_{\{\varphi\leq\psi-\varepsilon\}}\MA(\varphi),
so we obtain the desired estimate by letting ε → 0 \varepsilon\to 0
Original source context: S5.E1