When is a Set Measurable? | Real Analysis Part 1

Giuseppe Vitali constructed the first non-measurable set in 1905 by defining equivalence classes of real numbers differing by rationals, then selecting one representative from each class using the Axiom of Choice. This set V has no consistent measure: if it had measure zero, countably many copies would cover the unit interval with measure zero (contradiction); if it had positive measure, countably many copies would have infinite measure (also contradiction). Banach-Tarski pieces are similar non-measurable sets. Robert Solovay proved that without the Axiom of Choice, all subsets of reals are measurable. However, in ZFC, measurability is the exception—the well-behaved sets of applied mathematics are a special case within a sea of pathological objects that the Axiom of Choice guarantees but that no physical process can instantiate.

Vitali made infinite copies of his set, each shifted by a different rational number between -1 and 1. The union of these sets covers every real number between 0 and 1, but the total size must be between 1 and 3. However, if the Vitali set had size 0, adding it infinitely many times would still give 0. If it had any positive size, adding it infinitely many times would give infinity, not a value between 1 and 3. This contradiction shows the Vitali set is non-measurable. In 1924, Stefan Banach and Alfred Tarski used the Axiom of Choice to prove you could take a single solid ball, split it into just five pieces, and by carefully rotating and moving those pieces, reassemble them into two identical balls. This could be repeated to create infinite balls from one.

In 1905, Giuseppe Vitali constructed a set of real numbers between 0 and 1 by partitioning them into equivalence classes where two numbers are equivalent if their difference is rational. Using the Axiom of Choice, one selects exactly one representative from each class to form the Vitali set. This leads to a contradiction: if the Vitali set has measure zero, countably many copies (shifted by rational amounts) would have total measure zero. If it has positive measure, copies would have infinite total measure. However, the union covers [0,1], which has measure 1. This contradiction shows the Vitali set is non-measurable - its length cannot be defined.

Giuseppe Vitali discovered sets of numbers that cannot be measured. He considered the segment between 0 and 1, created copies of it, and took one point from each copy. This set produced contradictions when trying to measure it—getting different answers depending on how you look at it. This showed that some sets cannot be assigned a sensible measure.

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.

Lp spaces are collections of measurable functions defined on an abstract measure space, where functions are grouped into equivalence classes under the relation of equality almost everywhere; the p-norm is defined as the integral of |f|^p raised to the power 1/p, and for p=infinity, the space consists of essentially bounded functions with the essential supremum norm, which requires identifying functions that differ only on sets of measure zero to ensure the norm satisfies the property that ||f||=0 implies f=0.

For a measure space with finite measure, if 1 ≤ p < q ≤ ∞, then Lq ⊆ Lp. This means functions in Lq are automatically in Lp. The inclusion follows from Hölder's inequality: ||f||_p ≤ μ(E)^(1/p - 1/q) ||f||_q. The constant depends on the measure of the space. For infinite measure spaces, this inclusion does not hold in general.

Lp spaces are metric spaces consisting of all sequences of real or complex numbers where the sum of the absolute values raised to the power p converges, with the distance between two sequences defined as the p-th root of the sum of the absolute differences raised to the power p; these spaces satisfy the metric space axioms including non-negativity, identity of indiscernibles, symmetry, and the triangle inequality (proven via Minkowski's inequality), with L∞ space being a special case where p approaches infinity and distance is measured by the supremum norm.

Lp spaces are mathematical spaces consisting of all sequences where the sum of |x_i|^p is finite. For each value of p chosen, there exists a corresponding Lp space. When p=1, L1 consists of all sequences where the sum of absolute values is finite. When p=2, L2 consists of all sequences where the sum of squares is finite. Each Lp space can be equipped with a norm to become a normed space.

This comprehensive segment traces the complete development from intuitive vector norms to sophisticated LP spaces in functional analysis. Beginning with three concrete norms in n-dimensional space—Euclidean (circle unit sphere), taxi cab (square unit sphere), and maximum (diamond unit sphere)—the presentation introduces p-norms as a unifying framework. The critical insight emerges by recognizing vectors as functions on discrete measure spaces with counting measure, enabling generalization to arbitrary measure spaces. The technical machinery develops through convexity arguments, Young's inequality (ab ≤ aᴾ/P + b^Q/Q for conjugate exponents), and Hölder's inequality (∫|fg| ≤ ||f||_p × ||g||_q). These tools establish that p-norms satisfy triangle inequality, confirming they define valid norms. The quotient construction resolves the seminorm issue by identifying functions equal almost everywhere. Finally, LP spaces emerge as Banach spaces (complete with respect to p-norm), with dual spaces characterized for sigma-finite measures and p ≠ ∞. The special case p = 2 yields Hilbert spaces with inner products, providing the richest structure for function analysis through orthogonality and spectral methods.