By Tao T.
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The invention of limitless items by means of Wallis and endless sequence by way of Newton marked the start of the fashionable mathematical period. It allowed Newton to unravel the matter of discovering components less than curves outlined by means of algebraic equations, an success past the scope of the sooner tools of Torricelli, Fermat, and Pascal.
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If two signed measures µ, ν can be supported on disjoint sets, we say that they are mutually singular (or that µ is singular with respect to ν) and write µ ⊥ ν. 2. 5 (Jordan decomposition theorem). Every signed measure µ an be uniquely decomposed as µ = µ+ − µ− , where µ+ , µ− are mutually singular unsigned measures. ) We refer to µ+ , µ− as the positive and negative parts (or positive and negative variations) of µ. This is of course analogous to the decomposition f = f+ − f− of a function into positive and negative parts.
Real analysis (ii) (Homogeneity) cf c. Lp = |c| f Lp for all complex numbers (iii) ((Quasi-)triangle inequality) We have f +g Lp ≤ C( f Lp + g Lp ) for some constant C depending on p. If p ≥ 1, then we can take C = 1 (this fact is also known as Minkowski’s inequality). Proof. The claims (i), (ii) are obvious. 16) and is left as an exercise. 16). By the non-degeneracy property we may take f Lp and g Lp to be non-zero. Using the homogeneity, we can normalise f Lp + g Lp to equal 1, thus (by homogeneity again) we can write f = (1 − θ)F and g = θG for some 0 < θ < 1 and F, G ∈ Lp with F Lp = G Lp = 1.
We will not take this approach here, but see for instance [LiLo2000] for a discussion. 15. 14 below) in the special case when f , g are generalised step functions, say f = A1E and g = B1F with A, B non-zero. 24) µ(E ∩ F )1/r ≤ µ(E)1/p µ(F )1/q which can be easily deduced from the hypothesis p1 + 1q = 1r and the trivial inequalities µ(E ∩ F ) ≤ µ(E) and µ(E ∩ F ) ≤ µ(F ). 24) only holds if µ(E ∩ F ) = µ(E) = µ(F ), or in other words if E and F agree almost everywhere. Note the above computations also explain why the condition p1 + 1q = 1r is necessary.