Advanced Analysis (Set Theory and Measure Theory) MSc Notes

Advanced Analysis (Set Theory and Measure Theory) MSc Notes

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It is a fundamental part of modern mathematics and serves as the foundation for many other mathematical disciplines.

Set:

  • Definition: A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects within a set are called elements.

Element:

  • Definition: An element is an object that belongs to a particular set.
  • Notation: If (a) is an element of set (A), it is written as (a \in A).

Subset:

  • Definition: If every element of set (A) is also an element of set (B), then (A) is a subset of (B).

Union:

  • Definition: The union of two sets (A) and (B) is the set of all elements that are in (A), or (B), or in both.

Intersection:

  • Definition: The intersection of two sets (A) and (B) is the set of all elements that are common to both (A) and (B).

Complement:

  • Definition: The complement of a set (A) with respect to a universal set (U) is the set of all elements in (U) that are not in (A).
  • Notation: (A’) or (\bar{A}).

Cardinality:

  • Definition: The cardinality of a set is the number of elements in the set.
  • Notation: |A| = n.

Power Set:

  • Definition: The power set of a set (A) is the set of all possible subsets of (A), including the empty set and (A) itself.
  • Notation: \mathcal{P}(A) or (2^A).

Set theory provides a rigorous framework for defining and manipulating mathematical objects.

Measure theory is a branch of mathematics that extends the concepts of length, area, and volume from classical geometry to more general sets. It provides a rigorous foundation for the study of integration and probability theory.

Measurable Space:

  • Definition: A measurable space consists of a set (X) and a sigma-algebra (\mathcal{F}) of subsets of (X). The elements of (\mathcal{F}) are considered measurable sets.

Sigma-Algebra:

  • Definition: A sigma-algebra (\mathcal{F}) on a set (X) is a collection of subsets of (X) that satisfies certain properties:
    • (X) belongs to (\mathcal{F}).
    • If (A) is in (\mathcal{F}), then the complement of (A) is also in (\mathcal{F}).
    • If (A_1, A_2, \ldots) is a countable sequence of sets in (\mathcal{F}), then their union is also in (\mathcal{F}).

Measure:

  • Definition: A measure on a measurable space ((X, \mathcal{F})) is a function (\mu: \mathcal{F} \to [0, \infty]) that assigns a non-negative real number to each measurable set, satisfying the following properties:
    • (\mu(\emptyset) = 0),
    • If (A_1, A_2, \ldots) are pairwise disjoint sets in (\mathcal{F}), then (\mu(\cup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i)).

Lebesgue Measure:

  • Definition: The Lebesgue measure is a measure on the sigma-algebra of Lebesgue measurable sets in (\mathbb{R}^n). It extends the concept of length, area, and volume to more general sets.

Lebesgue Integral:

  • Definition: Given a measurable function (f) on a measurable space ((X, \mathcal{F})), the Lebesgue integral of (f) with respect to a measure (\mu) is denoted as (\int_X f \, d\mu). It generalizes the concept of integration from calculus.

Probability Measure:

  • Definition: A probability measure on a measurable space ((\Omega, \mathcal{F})) is a measure (\mathbb{P}: \mathcal{F} \to [0, 1]) such that (\mathbb{P}(\Omega) = 1).

Measure theory plays a crucial role in advanced analysis, functional analysis, and probability theory. It provides a solid mathematical foundation for understanding and working with more abstract and generalized notions of size and integration.

Advanced Analysis (Set Theory and Measure Theory) MSc Notes

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