The art of Function Design Measure and rKhs Tan xiaoyang 3.3,2011
The Art of Function Design -Measure and RKHS Tan xiaoyang 3.3，2011
topic Tell the story of measure(not measurement a brief account of rkhs. if we have time
topic • Tell the story of measure (not measurement!) • A brief account of RKHS, If we have time
Measure A probability space is a triple( S2, B. P)where S is the sample space corresponding to outcomes of some(perhaps hypo thetical) experiment e B is the o-algebra of subsets of 32. These subsets are called events P is a probability measure; that is, P is a function with domain B and range Definition 1.2.1: Let Q be a nonempty set and be an algebra on Q Then, a set function u on F is called a measure (a)pu(A)∈0,∞] for all a∈J; (b)p()=0: (c) for any disjoint collection of sets A1,A2,…,∈ F with U21An∈, (∪A)=∑An
Measure
Motivation of measure Problems of Riemann Integral 1. ask too much for a function - should be continous everywhere in general 2. even if it can be integrable the limit of a sequence of integral of functions may not be equal to the integral of the limit of function sequence Why?-the function value of dx may be unstable (See next slides
Motivation of Measure • Problems of Riemann Integral • 1. ask too much for a function – should be continous everywhere in general • 2. even if it can be integrable, the limit of a sequence of integral of functions may not be equal to the integral of the limit of function sequence. • Why? – the function value of dx may be unstable • (See next slides)
Motivation of measure The idea of Lebesgue is very simple not do the integrate by partition the domain but partition the codomain yn}--- y S=∑f()(x;-x;-1) 8=∑5m(E)
Motivation of Measure • The idea of Lebesgue is very simple • - not do the integrate by partition the domain, but partition the codomain