We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.
翻译:我们通过表明Prokhorov指标的有效趋同相当于有效的趋同,扩大实际措施趋同薄弱的有效框架,表明Prokhorov指标的有效趋同相当于实际趋同的弱弱;此外,我们为研究措施的模糊趋同的有效理论建立了一个框架;我们引入了一个统一的概念和非统一的概念,模糊的趋同,我们表明这两个概念是等同的;然而,有效模糊的趋同下的限制即使有限度也不可计算;我们举了一个有限、不可计算、有效的模糊的限制措施的例子,我们提供了必要和充分的条件,使有效的模糊趋同产生一个可计算的限制;最后,我们确定了一个充分的条件,使措施的有效弱和模糊的趋同相符合;作为必然结果,我们获得了一种有效的模式,即典型的弱点与概率措施的模糊趋同是等同。