# An epsilon of room: pages from year three of a mathematical by Tao T.

By Tao T.

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Real analysis Lebesgue measure on a line is singular continuous with respect to Lebesgue measure on a plane containing that line. 8. Suppose one is decomposing a measure µ on a Euclidean space Rd with respect to Lebesgue measure m on that space. Very roughly speaking, a measure is pure point if it is supported on a 0-dimensional subset of Rd , it is absolutely continuous if its support is spread out on a full dimensional subset, and is singular continuous if it is supported on some set of dimension intermediate between 0 and d.

Indeed, we shall see shortly that the two theorems are essentially equivalent to each other. 5, once we introduce the notion of a dual space. 17 (Continuity is equivalent to boundedness for linear operators). Let T : X → Y be a linear transformation from one normed vector space (X, X ) to another (Y, Y ). Then the following are equivalent: (i) T is continuous. (ii) T is continuous at 0. (iii) There exists a constant C such that T x x ∈ X. Y ≤C x X for all Proof. It is clear that (i) implies (ii), and that (iii) implies (ii).

In particular, this implies that a non-trivial measure that is singular with respect to m cannot be expressed in the form mf . 4 (Lebesgue-Radon-Nikodym theorem). Let m be an unsigned σ-finite measure, and let µ be a signed σ-finite measure. Then there exists a unique decomposition µ = mf + µs , where f ∈ L1 (X, dm) and µs ⊥ m. If µ is unsigned, then f and µs are also. Proof. We prove this only for the case when µ, ν are finite rather than σ-finite, and leave the general case as an exercise. 2 and the previous observation that mf cannot be mutually singular with m for any non-zero f , so it suffices to prove existence.