Measure theory = 测度论 /

introduction
o. conventions and notation
1.notation: euclidean space
2.operations involving
3.inequalities and inclusions
4.a space and its subsets
5.notation: generation of classes of sets
6.product sets
7.dot notation for an index set
8.notation: sets defined by conditions on functions
9.notation: open and closed sets
10.limit of a function at a point
11.metric spaces
12.standard metric space theorems
13.pseudometric spaces
i. operations on sets
i.unions and intersections
2.the symmetric difference operator
3.limit operations on set sequences
4.probabilistic interpretation of sets and operations on them
.ii. classes of subsets of a space
1.set algebras
2.examples
3.the generation of set algebras
4.the borel sets of a metric space
5.products of set algebras
6.monotone classes of sets
iii. set functions
1.set function definitions
2.extension of a finitely additive set function
3.products of set functions
4.heuristics on a algebras and integration
5.measures and integrals on a countable space
6.independence and conditional probability (preliminary discussion)
7.dependence examples
8.inferior and superior limits of sequences of measurable sets
9.mathematical counterparts of coin tossing
10.setwise convergence of measure sequences
11.outer measure
12.outer measures of countable subsets of r
13.distance on a set algebra defined by a subadditive set function
14.the pseudometric space defined by an outer measure
15.nonadditive set functions
iv. measure spaces
1.completion of a measure space (s,$,~,)
2.generalization of length on r
3.a general extension problem
4.extension of a measure defined on a set algebra
5.application to borel measures
6.strengthening of theorem 5 when the metric space s is complete and separable
7.continuity properties of monotone functions
8.the correspondence between monotone increasing functions on r and measures on b(r)
9.discrete and continuous distributions on r
10.lebesgue-stieltjes measures on r/v and their corresponding monotone functions
11.product measures
12.examples of measures on rn
13.marginal measures
14.coin tossing
15.the caratheodory measurability criterion
16.measure hulls
v. measurable functions
1.function measurability
2.function measurability properties
3.measurability and sequential convergence
4.baire functions
5.joint distributions
6.measures on function (coordinate) space
……
vi. intezration
vii. hilbert soace
viii. convergence of measure sequences
ix. signed measures
x. measures and functions of bounded variation on r
xi. conditional expectations; martingale theory
notation
index