Authors: Johan Aspegren
In this article we will prove a version of the Beppo Levi`s lemma for the complex valued functions. This achieved by making a more stronger asumption that is assumed in Beppo Levi`s lemma. We will assume that the sum of measurable functions that is absolutely convergent almost everywhere is integrable. We will prove that it implies the asumptions of the Beppo Levi lemma, if we consider functions that are non-negative. It can be argued that our version is more suitable to applications, and we will prove a new probability law. We will show that with our asumptions in probability theory it follows that the expected value is countable additive. Moreover, it follows that in strong law of large numbers we don`t need to make any asumptions on distributions and the mean of the sample will convergence almost surely to the mean of the expected values.
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