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Lebesgue’s Dominated Convergence Theorem: Let $$(f_{i})_{i=1}^{\infty}$$ be measurable on $$E$$ and $$g$$ be non-negative and integrable on $$E$$ such that $$g$$ dominates each $$f_{i}$$ almost everywhere on $$E$$, i.e., for each $$i$$, $$|f_{i}(x)|\leq g(x)$$ for almost all $$x\in E$$. Then, if $$(f_{i})_{i=1}^{\infty}$$ converges almost everywhere on $$E$$ to $$f$$, $$f$$ is integrable on $$E$$ and $$\lim_{i\to\infty}\int_{E}\mathrm{f_{i}}=\int_{E}\mathrm{f}$$.

The Basel Sum: $$\sum_{i=1}^{\infty}\frac{1}{i^{2}}=\frac{Pi^{2}}{6}$$

Citing other papers is easy. Voilà: (CMS/CERN 2012) or (Holstein 2009). Click on the cite button in the toolbar to search articles and cite them. Authorea also comes with a powerful commenting system. Don’t agree that $$E=mc^{3}$$?!? Highlight the text you want to discuss or click the comment button. Find out more about using Authorea on our help page.

### References

1. CMS/CERN. A New Boson with a Mass of 125 GeV Observed with the CMS Experiment at the Large Hadron Collider. Science 338, 1569-1575 (2012). Link

2. Barry R Holstein. The mysterious disappearance of Ettore Majorana. J. Phys.: Conf. Ser. 173, 012019 IOP Publishing, 2009. Link