1.2 Basic concepts: sets, functions, and measures

Set theory, functions, and measures form the foundation of measure theory. These concepts provide the tools to analyze and quantify complex mathematical structures. Understanding their properties and relationships is crucial for grasping more advanced topics in this field.

Measures extend the notion of length, area, and volume to more abstract spaces. They allow us to assign sizes to sets in a consistent way, respecting key properties like non-negativity and additivity. This framework enables the development of powerful integration techniques and probability theory.

Set Operations and Manipulation

Set Theory Fundamentals

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• A set is a collection of distinct objects
• Set theory provides a foundation for mathematical analysis and measure theory
• The universal set contains all elements under consideration in a given context

Basic Set Operations

• The union of two sets A and B, denoted $A \cup B$, contains all elements that belong to either A or B, or both
  • Example: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cup B = \{1, 2, 3, 4, 5\}$
• The intersection of two sets A and B, denoted $A \cap B$, contains all elements that belong to both A and B
  • Example: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cap B = \{3\}$
• The complement of a set A, denoted $A^c$ or $A'$, contains all elements in the universal set that do not belong to A
  • Example: If the universal set is $\{1, 2, 3, 4, 5\}$ and $A = \{1, 2, 3\}$, then $A^c = \{4, 5\}$
• The difference of two sets A and B, denoted $A \setminus B$, contains elements in A that are not in B
  • Example: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \setminus B = \{1, 2\}$
• The symmetric difference of two sets A and B, denoted $A \triangle B$, contains elements that belong to either A or B, but not both
  • Example: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \triangle B = \{1, 2, 4, 5\}$

Advanced Set Relationships

• De Morgan's laws describe the relationship between set operations and their complements
  • $(A \cup B)^c = A^c \cap B^c$: The complement of the union is the intersection of the complements
  • $(A \cap B)^c = A^c \cup B^c$: The complement of the intersection is the union of the complements
• Power set: The power set of a set A, denoted $\mathcal{P}(A)$, is the set of all subsets of A, including the empty set and A itself
  • Example: If $A = \{1, 2\}$, then $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$
• Cartesian product: The Cartesian product of two sets A and B, denoted $A \times B$, is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$
  • Example: If $A = \{1, 2\}$ and $B = \{x, y\}$, then $A \times B = \{(1, x), (1, y), (2, x), (2, y)\}$

Function Properties and Types

Function Fundamentals

• A function $f$ from a set X to a set Y, denoted $f: X \to Y$, assigns to each element $x$ in X a unique element $f(x)$ in Y
• The domain of a function $f$ is the set of all input values $(x)$ for which the function is defined
• The codomain of a function $f$ is the set Y that contains all possible output values
• The range or image of a function $f$ is the set of all output values $(f(x))$ that the function actually attains

Injectivity, Surjectivity, and Bijectivity

• A function $f: X \to Y$ is injective (one-to-one) if for any two distinct elements $x_1$ and $x_2$ in X, $f(x_1) \neq f(x_2)$ in Y
  • Example: $f(x) = 2x$ is injective because each output corresponds to a unique input
• A function $f: X \to Y$ is surjective (onto) if for every element $y$ in Y, there exists at least one element $x$ in X such that $f(x) = y$
  • Example: $f(x) = x^2$ is surjective on the domain $\mathbb{R}$ and codomain $[0, \infty)$ because every non-negative real number has a square root
• A function that is both injective and surjective is called bijective (one-to-one correspondence)
  • Example: $f(x) = 2x + 1$ is bijective on the domain and codomain $\mathbb{R}$

Measurable Functions

• A function $f: X \to Y$ is measurable if the preimage of any measurable set in Y is a measurable set in X
• Measurable functions are important in measure theory and integration
• Continuous functions and step functions are examples of measurable functions

Measures and their Properties

Measure Spaces

• A measure space is a triple $(X, \Sigma, \mu)$, where X is a set, $\Sigma$ is a $\sigma$-algebra of subsets of X, and $\mu$ is a measure on $\Sigma$
• A $\sigma$-algebra $\Sigma$ on a set X is a collection of subsets of X that includes X itself, is closed under complement, and is closed under countable unions
  • Example: The Borel $\sigma$-algebra on $\mathbb{R}$ is the smallest $\sigma$-algebra containing all open intervals
• A set A is said to be measurable if it belongs to the $\sigma$-algebra $\Sigma$

Measure Properties

• A measure $\mu$ on a $\sigma$-algebra $\Sigma$ is a function $\mu: \Sigma \to [0, \infty]$ that satisfies:
  • Non-negativity: For any set A in $\Sigma$, $\mu(A) \geq 0$
  • Null empty set: $\mu(\emptyset) = 0$
  • Countable additivity: For any countable collection $\{A_n\}$ of pairwise disjoint sets in $\Sigma$, $\mu(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu(A_n)$
• The Lebesgue measure on $\mathbb{R}^n$ is a fundamental example of a measure, assigning the conventional length, area, or volume to suitable subsets
  • Example: The Lebesgue measure of an interval $[a, b]$ is its length $b - a$
• Other examples of measures include probability measures, counting measures, and the Hausdorff measure