On a measurable space, let $m_n, n=1, \dots, $ be a sequence measures on it and $\mu$ is another measure.
let $Z := \{ϕ∈L^2(\mu): ∫ ϕ\,d\mu = 0 \}$.
The first type of convergence of $m_i$'s is defined as $ \lim_{n \to \infty} \sup_{\phi\in Z:\,\|\phi\|_2=1} \int \phi \, dm_n =0. $ This is motivated by the definition of a stochastic process being ρ-mixing.
The second type of convergence of $m_i$'s is defined as $ \lim_{n \to \infty} \sup_{\phi\in Z:\,\|\phi\|_\infty=1} \int \phi \, dm_n =0. $ This is motivated by the definition of a stochastic process being α-mixing.
The third type of convergence of $m_i$'s is defined as $ \lim_{n\to \infty} \sup_{0\leq\phi\leq1} \Big| \int \phi \, dm_n - \int \phi \, dQ \Big| = 0 $ This is motivated by the definition of a stochastic process being β-mixing. A side question: what does $0\leq\phi\leq1$ really means?
Are there names for these types of convergence of measures? I would like to know about their relations with other types of convergence of a sequence of measures, such as weak and weak* convergence.
I have tried to read Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics 155 (2): 155–169. cited in the Wikipedia article for the three types of mixing, and also traced Mixing Conditions for Markov Chains Yu. A. Davydov Theory Probab. Appl. 18-2 (1974) cited in that paper for the definitions of β-mixing. But I failed to see answers to my above questions.
Thanks!