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We address the study of topologically invariant means and almost convergenceon the real numbers $\mathbb{R}$. Here, the former is a certain class ofinvariant means on $L^{\infty}(\mathbb{R})$ and the latter is a summabilitymethod defined by them. Almost convergence on $\mathbb{R}$ was firstlyintroduced by Raimi (1957) as a generalization of Lorentz's almost convergencefor bounded sequences. We extensively generalize his result of analyticcharacterization of almost convergence and explore its application to thetheory of Hardy space. Specifically, we establish the relation between theasymptotic behavior on the imaginary axis and that at infinity of boundedanalytic functions defined on the right half plane.

On Almost convergence on the real line and its application to bounded analytic functions