The Art of Function Design -Measure and RKHS

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

But For any function f(x), is m(Ei) always meaningful? 8=∑m(E) 区间有“长度”,但并不意味在复杂集上 有长度, or it is unclear how to define one for them 因此需要重新审査“长度”的概念。 长度、体积、重量。。。统称测度

But.. • For any function f(x), is m(Ei) always meaningful? • 区间有“长度”，但并不意味在复杂集上 总有长度，or it is unclear how to define one for them！ • 因此需要重新审查“长度”的概念。 长度、体积、重量。。。统称 测度

Defining measure It should be a nonegative set function Le, u: X->R+ Lots of functions likes these not all can be called measure:制定规则 ·规则1:空集的测度为0 规则2任意多个互不相交集合的总测度等于各子 测度之和( countable additivity) 这等于什么都没说, but check it before adding more rules

Defining Measure • It should be a nonegative set function – i.e., u: X->R+ • Lots of functions likes these, not all can be called measure: 制定规则 • 规则1: 空集的测度为0 • 规则2: 任意多个互不相交集合的总测度等于各子 测度之和 （countable additivity） • 这等于什么都没说，but check it before adding more rules!

Bad news Even for these two simple rules it will not work for the real line Change the rule? Lebesque said, no! lets restrict the domain to make it work and. since these sets in the new domain fit the definition we call them measurable sets · Your questions?

Bad news • Even for these two simple rules, it will not work for the real line! • Change the rule? • Lebesgue said, no! let’s restrict the domain to make it work, and, since these sets in the new domain fit the definition, we call them measurable sets. • Your questions?

questions how to restrict the domain to make the ules work? 2. even if the definition orks is it meaningful? Nol we we'l add the third rule the measure of any interval (a, b] on real line should be b-a this make the definition semantically works How about other complex sets? The measure machine will do it automatically for you. So how it works?

questions • 1. how to restrict the domain to make the rules work? • 2. even if the definition works, is it meaningful? – No! we we’ll add the third rule : the measure of any interval (a,b] on real line should be b-a. this make the definition semantically works. – How about other complex sets? – The measure machine will do it automatically for you. So how it works?

Building the system Step 1, let's give a set of all interval like(a, b] on the real line, named P, a measure m((a, b]=b-a Step 2, now we want the measure can be algebrally caculated finitely, so we will extend the domain from p to some ring R, and onR, we use the same m as our measure but now it can be finitely added, substracted But we want more How to do it infinitely?

Building the system • Step 1, let’s give a set of all interval like (a, b] on the real line, named P, a measure m((a,b])=b-a. • Step 2, now we want the measure can be algebrally caculated finitely, so we will extend the domain from P to some ring R, and on R, we use the same m as our measure but now, it can be finitely added, substracted,… But we want more, How to do it infinitely?