By Tao T.

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3 of Structure and Randomnessvwas a fully finitary analogue of the infinite convergence principle), though one can certainly form some weaker finitary statements that capture a portion of the strength of this theorem. 4 and Markov’s inequality, and has meaningful content even when the underlying space X is finite, but this type of decomposition is not as powerful as the full Lebesgue decompositions (mainly because the size of the support for µ>k is relatively large compared to the threshold k).

One often restricts attention to group actions that are linear (to preserve the vector space structure), continuous (to preserve topological structure), unitary (to preserve inner product structure), isometric (to preserve metric structure), and so forth. Besides giving us useful symmetries to spend, the presence of such group actions allows one to apply the powerful techniques of representation theory, Fourier analysis, and ergodic theory. However, as this is a foundational real analysis class, we will not discuss these important topics much here (and in fact will not deal with group actions much at all).

Topological structure. g. ) that one wishes to carefully distinguish from each other. ), it is important to know when certain subsets of V enjoy key topological properties (most notably compactness and connectedness), and to know which operations on V are continuous. For all of this, one needs a topology on V . If one already has a metric, then one of course has a topology generated by the open balls of that metric; but there are many important topologies on function spaces in analysis that do not arise from metrics.

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