Plan Jets of a Sum along a Common Realisation

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\varphi_{1},\varphi_{2}:\Sigma^{2}_{d}\to\mathbb{R}$, and let $\varphi_{1}+\varphi_{2}:\Sigma^{2}_{d}\to\mathbb{R}$ be given by $(\varphi_{1}+\varphi_{2})(\nu)=\varphi_{1}(\nu)+\varphi_{2}(\nu)$. Let $\lambda\in\Sigma^{2}_{d}$, let $\delta_{1},\delta_{2}\ge0$ be real, let $(H,M,\Omega)$ be a tracial W*-probability space, and let $X,P_{1},P_{2}$ be $L^{2}$ $d$-tuples of $(H,M,\Omega)$ with $\mathrm{law}(X)=\lambda$. The sum $P_{1}+P_{2}$ is that of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; pairs and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $J^{+}_{\delta}$ and $J^{-}_{\delta}$ are the plan superjet and the plan subjet with slack $\delta$.

1. (Superjets)¶ If $\mathrm{law}(X,P_{i})\in J^{+}_{\delta_{i}}\varphi_{i}(\lambda)$ for $i=1,2$, then $\mathrm{law}(X,P_{1}+P_{2})\in J^{+}_{\delta_{1}+\delta_{2}}(\varphi_{1}+\varphi_{2})(\lambda)$.

2. (Subjets)¶ If $\mathrm{law}(X,P_{i})\in J^{-}_{\delta_{i}}\varphi_{i}(\lambda)$ for $i=1,2$, then $\mathrm{law}(X,P_{1}+P_{2})\in J^{-}_{\delta_{1}+\delta_{2}}(\varphi_{1}+\varphi_{2})(\lambda)$.