Measure Theory 1: Sigma Algebras Explained | Mathematics Lecture

A Sigma algebra on a set Ω is a collection of subsets containing Ω, closed under complements and countable unions. The Borel sigma algebra B(ℝ) is the minimal sigma algebra containing all intervals (-∞, x] for real x. Elements are called Borel sets. Examples include: half-open intervals (-∞, x) as countable unions of closed intervals; closed intervals [x, y] as differences of half-open intervals; open intervals (x, y) as differences; single points and countable sets (ℕ, ℤ, ℚ) as countable unions; and ℝ itself as a countable union of (-∞, n]. Despite containing many important sets, B(ℝ) is not the power set of ℝ—there exist non-Borel subsets.

The Borel sigma-algebra on an interval [a,b] is the smallest collection of subsets that contains all closed bounded intervals and is closed under countable union, countable intersection, and complementation; this collection, denoted as script B, includes all open sets, semiopen sets, countable unions of intervals, the set of all rationals in [a,b], and the set of all irrationals in [a,b], but does not contain all possible subsets of [a,b].

The Borel sigma algebra, denoted B(X), is the sigma algebra generated by the open sets of a topological space X. It is defined as σ(T), where T is the topology (collection of open sets). In metric spaces like ℝ^n, it uses the standard topology. The Borel sigma algebra includes all open sets and is closed under countable unions and complements. It is not the power set (which would include all possible subsets), but it is the most suitable sigma algebra for measure theory because it contains all sets we want to measure while being large enough to satisfy meaningful measure properties.

The Borel sigma algebra, denoted by B, is defined as the intersection of all sigma algebras that contain every open set; since it contains all open sets and is closed under countable unions, intersections, and complements, it follows that every interval is a Borel set.

In measure theory, a function f: Ω → ℝ̄ is measurable if the preimage of every Borel set belongs to the σ-algebra F; equivalently, it suffices to check that f⁻¹((−∞, x]) ∈ F for all x ∈ ℝ. Simple functions are measurable functions that take finitely many values and can be expressed as finite linear combinations of indicator functions of measurable sets. For a non-negative simple function f = Σc_k·1_E_k, its integral is defined as Σc_k·μ(E_k), and this definition extends to all non-negative measurable functions via monotone convergence, and finally to all measurable functions by decomposing them into positive and negative parts.

A function f: X → ℝ is called measurable if for every real number α, the set f⁻¹([α, ∞)) is a measurable set; this definition extends to extended real-valued functions, and it can be shown that a function is measurable if and only if any one of the four equivalent conditions holds: f⁻¹((α, ∞)), f⁻¹([α, ∞)), f⁻¹((−∞, α)), or f⁻¹((−∞, α]) is measurable for all α ∈ ℝ, and measurable functions are closed under countable unions, intersections, and complements.

This section covers measurable functions: (1) f is measurable if {x: f(x)>a} is measurable for all a∈R; (2) Translation invariance: if E is measurable, E+a is measurable; (3) Algebra of measurable functions: if f,g measurable, then f+g, f-g, f×g, f/g (g≠0) are measurable; (4) Limits: lim sup f_n, lim inf f_n, and lim f_n (if exists) are measurable. These properties ensure measurability is preserved under standard operations.

Measurable functions generalize continuous functions by preserving measurable structure rather than topological structure. A function f: (E, A) → (F, B) is measurable if f⁻¹(B) ∈ A for all B ∈ B. Continuous functions are always Borel measurable because they preserve open sets, which generate the Borel tribe. The composition of measurable functions remains measurable, as (g∘f)⁻¹(C) = f⁻¹(g⁻¹(C)) preserves measurability through composition.

If f is continuous and g is measurable, then f ∘ g is measurable. A function f is measurable if and only if for every open set G, the inverse image f⁻¹(G) is measurable. If f is measurable on each Eᵢ in a countable collection of disjoint measurable sets, then f is measurable on ∪Eᵢ. If f is measurable on a measurable set E, and E₁ is a measurable subset of E, then f is measurable on E₁. If f and g are measurable functions on a measurable set E, then the set {x ∈ E : f(x) < g(x)} is measurable, as it equals ∪_{r ∈ ℚ} ({x : f(x) < r} ∩ {x : g(x) > r}), a countable union of measurable sets.

The Lebesgue integral, developed by Henri Lebesgue in 1902, revolutionized integration by shifting from partitioning the x-axis (as in the Riemann integral) to partitioning the y-axis based on function values, combined with the Lebesgue measure which assigns length zero to individual points; this approach successfully integrates functions like the Dirichlet function (which equals 1 at rationals and 0 at irrationals) that the Riemann integral cannot handle, and forms the foundation for modern analysis, probability theory, Fourier analysis, and quantum mechanics through L2 Hilbert spaces.

The Dirichlet function D(x) = 1 if x is rational, 0 if x is irrational, has Lebesgue integral 0 from 0 to 1. This is because the set of rational numbers has Lebesgue measure 0. The Lebesgue integral ignores values on sets of measure 0, so changing D(x) on rational points (which form a set of measure 0) does not change the integral. This illustrates how Lebesgue integration differs from Riemann integration, which would not even exist for this function.

The Lebesgue integral extends integration to functions that are discontinuous everywhere, such as the indicator function for rational numbers (1 on rationals, 0 on irrationals). Unlike the Riemann integral which partitions the domain and creates vertical rectangles, the Lebesgue integral partitions the range and creates horizontal rectangles. This requires defining a measure on sets, which leads to measure theory. The Lebesgue measure assigns length to subsets of the real line. Single points have measure zero, and countable sets like the rationals also have measure zero. Therefore, the Lebesgue integral of the indicator function for rationals on [0,1] is 1 × 0 = 0, meaning a randomly chosen number has probability zero of being rational.

The Lebesgue integral is constructed in three steps: (1) Simple functions are measurable functions taking finite values, represented as sums of function values times indicator functions of disjoint sets; their integral is the sum of value × measure. (2) For non-negative measurable functions, the approximation lemma provides a sequence of simple functions approximating from below, and the integral is defined as the limit of these integrals. (3) For general measurable functions, decompose into positive and negative parts, then define the integral as ∫f⁺ - ∫f⁻, requiring both integrals to be finite.

This section establishes the axiomatic foundation of probability theory used throughout the course. A random experiment allows listing all possible outcomes before execution, forming the sample space Ω. Events are subsets of Ω. A sigma algebra F over Ω satisfies three conditions: contains empty set, closed under unions, and closed under complements. The probability space is a triplet (Ω, F, P) where P satisfies non-negativity, P(Ω)=1, and additivity for mutually exclusive events. Conditional probability P(B|A) = P(A∩B)/P(A) measures probability of B given A occurred. A random variable is a real-valued function X: Ω → R where inverse images of intervals belong to F. The largest sigma algebra makes any real-valued function a random variable, enabling mathematical modeling of uncertain phenomena.

The axiomatic approach to probability provides a more general framework than the classical definition, where probability is defined as a function P(E) that satisfies three axioms: (1) P(E) ≥ 0 for any event E, (2) P(S) = 1 where S is the sample space, and (3) for mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). This approach can be applied to both finite and infinite sample spaces, unlike the classical definition which requires equally likely outcomes. The video demonstrates this through various examples including coin tosses, dice rolls, card draws, and combinatorial problems, showing how to calculate probabilities using combinations and permutations when selecting multiple items.

The axiomatic definition of probability, developed by Russian mathematician A.N. Kolmogorov, establishes three fundamental axioms: (1) Probability is always non-negative (P(A) ≥ 0), (2) The probability of a certain event is 1 (P(S) = 1), and (3) For any finite number of pairwise mutually exclusive events, the probability of their union equals the sum of their individual probabilities (P(A1 ∪ A2 ∪ ... ∪ An) = P(A1) + P(A2) + ... + P(An)). These axioms provide a rigorous mathematical foundation for probability theory and enable the derivation of important results such as P(A') = 1 - P(A) and P(A' ∩ B') = P(A') - P(B) + P(AB).

This comprehensive section presents the mathematical foundations of probability and demonstrates practical applications. Probability is defined as a number between 0 and 1 for any event, with the certain event having probability 1 and the impossible event having probability 0. For incompatible events, P(A ∪ B) = P(A) + P(B). Since an event and its complement are incompatible and cover the entire sample space, P(A) + P(Ā) = 1, so P(Ā) = 1 - P(A). The classical probability formula states that when outcomes are equally likely, P(A) = number of favorable outcomes / total number of possible outcomes. The section demonstrates applications using a class of 30 students where 20 study English, 7 study Spanish, and 3 study both. The probability of selecting a student studying English is 20/30 = 2/3. The probability of studying at least one language uses the union formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 24/30 = 4/5. The section also covers geometric probability where probability is proportional to area, and probability with multiple dice rolls.

The axiomatic approach to probability defines probability through three fundamental axioms: (1) Probability of any event is between 0 and 1 inclusive, (2) Probability of the sample space equals 1, and (3) For mutually exclusive events, the probability of their union equals the sum of their individual probabilities. Key concepts include mutually exclusive events (where intersection probability is zero), complementary events (where P(A') = 1 - P(A)), and De Morgan's Laws for set operations. The probability of the union of two events is calculated as P(A ∪ B) = P(A) + P(B) - P(A ∩ B), which simplifies to P(A) + P(B) for mutually exclusive events.