Chapter 5 · Subchapter 1

Simple Functions and the First Integral

The average of a random variable is the course's central number. We define it honestly — from finite betting tickets upward through approximation — and the construction itself teaches the method every later proof will reuse.

In this subchapter

The problem with "average"
Simple functions: finite tickets
Nonnegative variables: climbing the staircase
The general definition and its algebra
Infinite expectations: the honest cases
First computations

5.1.1The problem with "average"

What should E[X], the average of a random variable, be? The finite-outcome answer is a weighted sum: Σ x_k P(X = x_k). But a uniform variable on [0,1] has no atoms — every single value has probability zero — so the weighted sum is empty of terms: it computes nothing. And the course's own monsters are worse: the Cauchy-tailed variables (5.5) have averages that do not exist, and singular laws like the Cantor distribution (4.2 §5) have no density to integrate against. "Just integrate x f(x) dx" quietly assumes away every interesting case.

Example 1 · Why atoms cannot carry the uniform variable

On ([0,1], ℒ, λ) let X(ω) = ω. For any point a, P(X = a) = λ({a}) = 0 (3.1 Example 2). A "sum over values, weighted by P(X = value)" gives zero contribution from every value and would total 0 — absurd, since X averages visibly 1/2 by symmetry. The mass is there but it is spread; the definition of expectation must talk to spread-out mass, and the only tool we own for that is the measure — the probability of sets, not of points. So the definition will be built from level sets: the events {X ≤ t} that measurability (4.1) guaranteed exist.

5.1.2Simple functions: finite tickets

Definition — simple function

A simple function takes finitely many values: Y = Σ_k=1ⁿ a_k 1_{A_k}, with a_k ∈ ℝ and A_k ∈ F pairwise disjoint. Read it as a betting ticket: "you receive a₁ if event A₁ happens, a₂ if A₂, …". The indicator 1_A — 1 on A, 0 off it — is the one-bit ticket; every simple function is a portfolio of one-bit tickets.

Definition — expectation of a simple function

For Y = Σ a_k 1_{A_k} as above,

E[Y] = Σ_k=1ⁿ a_k P(A_k)

the value-weighted sum of event probabilities. If some a_k are negative, split: write Y = Y⁺ − Y⁻ with Y^± ≥ 0, and require E[Y⁺] and E[Y⁻] not both infinite to avoid ∞ − ∞; for bounded simple functions this is automatic.

Example 2 · Tickets, priced

(a) A die roll's identity: E[X] = (1+2+⋯+6)/6 = 3.5 — the familiar arithmetic mean as the n = 6 ticket. (b) The banded uniform: cut [0,1] into the four quarters and let Y = 0.125 on [0,1/4), 0.375 on [1/4,1/2), 0.625 on [1/2,3/4), 0.875 on [3/4,1] (midpoint banding). Then E[Y] = (0.125+0.375+0.625+0.875)/4 = 0.5 — the banding approximates the uniform average exactly at the centroid level. (c) A digital option paying 1 if the (uniform) underlying ends above 0.4: E[1_(0.4,1]] = λ((0.4,1]) = 0.6. Note what (c) teaches: the expectation of an indicator is a probability — E[1_A] = P(A) — so the theory of expectation contains all of probability theory; nothing is lost by moving from events to functions.

5.1.3Nonnegative variables: climbing the staircase

Now the idea that defines the Lebesgue theory of integration, in one sentence: to integrate a general nonnegative function, approximate it from below by simple functions and take the limit of the ticket prices. The construction is the banding you already saw twice — in the proof of Doob–Dynkin (4.3) and Theorem 2's limsup step of 4.1.

Definition — expectation of a nonnegative variable

For X ≥ 0, let Y_n be the n-th banded approximation: cut the range into bands of height 2⁻ⁿ up to level n, and set Y_n(ω) = (k−1)2⁻ⁿ where X's value falls in the k-th band, 0 above level n (truncation). Explicitly:

Y_n = Σ_k=1^n·2ⁿ (k−1)2⁻ⁿ · 1_{{ (k−1)2⁻ⁿ ≤ X < k·2⁻ⁿ }}

Each Y_n is simple (its level sets are in F by 4.1 Theorem 2), and Y_n ↑ X pointwise: finer bands and higher truncation creep up to X from below, never overshooting. Then define

E[X] := lim_n→∞ E[Y_n] ∈ [0, ∞]

The limit exists (monotone sequence of reals), and — the nontrivial fact, Theorem 1 — it does not depend on the approximating sequence.

Theorem 1 — the definition is well-posed

If 0 ≤ Y₁ ≤ Y₂ ≤ ⋯ are simple functions with Y_n ↑ X pointwise, then lim_n E[Y_n] is the same for every such sequence. So "expectation" is a property of X, not of the approximation.

Proof — the zigzag trade, the course's first real ε-management

Fix two staircases: Y_n ↑ X and Z_m ↑ X, both simple. Claim: lim_m E[Z_m] ≥ lim_n E[Y_n]; by symmetry the reverse holds too, and we are done.

Fix n and a tolerance: consider the event A_m = {Y_n ≤ Z_m + ε} — hmm, cleaner: the event B_m = {Z_m > Y_n − ε}. Since Z_m ↑ X and Y_n < X + δ fails only where X = Y_n... let us state the standard zigzag directly. Fix n, fix ε > 0, and let C_m = {Y_n ≤ Z_m + ε}. On any ω, Y_n(ω) < X(ω) + ε is automatic for large enough m? No — the roles matter. Since Z_m(ω) ↑ X(ω) ≥ Y_n(ω), every ω eventually lands in C_m: so C_m ↑ Ω, and continuity from below (2.2 Theorem 2) gives P(C_m) → 1.

Now the trade. Write Z_m = Z_m·1_{C_m} + Z_m·1_{C_m^c}. On C_m, Z_m ≥ Y_n − ε; on the complement, keep only nonnegativity: Z_m ≥ 0. So

E[Z_m] ≥ (Y_n − ε)-part: ≥ E[Y_n·1_{C_m}] − ε ≥ E[Y_n] − ‖Y_n‖_∞·P(C_m^c) − ε

using linearity on simple functions (a finite check), the bound on C_m, and the tail estimate E[Y_n·1_{C_m^c}] = E[Y_n] − E[Y_n·1_{C_m}]. As m → ∞, P(C_m^c) → 0, and ‖Y_n‖_∞ (the largest value Y_n takes — a maximum, since Y_n is simple) is a finite number. So lim_m E[Z_m] ≥ E[Y_n] − ε. Let ε ↓ 0, then n → ∞: lim_m E[Z_m] ≥ lim_n E[Y_n]. Symmetry closes it. ∎

What to take from the proof

Three moves, all reusable: (1) the continuity theorem converts pointwise convergence into probability convergence of the "good set"; (2) the split — bound the approximation on the good set, discard the bad set at cost ‖·‖_∞ · P(bad) — is the ancestor of every truncation argument in the course; (3) tolerances ε enter and leave: never begin a proof with ε you are not prepared to send to zero at the end. This is the ε-management the course promised to teach slowly (1.1's brittleness-flagging, now in action).

5.1.4The general definition and its algebra

Definition — expectation (general case)

For any random variable X, split X = X⁺ − X⁻ where X⁺ = max(X, 0) and X⁻ = max(−X, 0) (both nonnegative, both random variables by 4.1 Theorem 2). Define E[X] = E[X⁺] − E[X⁻] whenever at least one of the two terms is finite; if both are infinite, E[X] does not exist. If E[X⁺] < ∞ and E[X⁻] < ∞, X is integrable and we write X ∈ L¹.

Theorem 2 — algebra of expectation

On the class where the terms exist: E[aX + bY] = aE[X] + bE[Y] (linearity); if X ≤ Y everywhere then E[X] ≤ E[Y] (monotonicity); |E[X]| ≤ E|X| (triangle); and for X with a bounded range, |E[X]| ≤ ‖X‖_∞.

Proof sketch — inherit, then approximate

Each rule is a finite computation for simple functions (linearity: expand both sides as sums over the common refinement of the two partitions — the same refinement move as 2.3's common coarsening, run backwards). For nonnegative X, Y: band both with the same grid, inherit the rule at each n, and pass to the limit — the limit of an inequality is an inequality, so monotonicity survives; linearity needs the zigzag theorem (Theorem 1) to know both sides approximate coherently. The general case splits into ± parts. ∎

5.1.5Infinite expectations: the honest cases

Example 3 · A variable with E[X] = ∞, on a perfectly legal space

On ([0,1], ℒ, λ) let X(ω) = 1/ω for ω > 0, and X(0) = 0. First, a calibration that sharpens the point: the lighter singularity ω^−1/2 has E[ω^−1/2] = ∫₀¹ ω^−1/2 dω = [2√ω]₀¹ = 2, finite — a spike at a single point is harmless. But for 1/ω, price the truncations: X ∧ n (X capped at n) is a simple-function limit, and its expectation is

E[X ∧ n] = ∫₀¹ min(1/ω, n) dω = ∫₀^1/n n dω + ∫_1/n¹ dω/ω = 1 + ln n → ∞

so E[1/ω] = ∞: the variable is finite at every single outcome, yet its expectation is infinite. Note also the harmonic-series form: banding at height 1/k charges λ({X ≥ k}) = 1/k, and the price is the divergent harmonic series — first-year calculus divergence, promoted to a probability statement. Expectation is not a property of typical outcomes; it is a property of the tail, and tails are exactly where intuition underestimates. 5.5 builds a whole lesson on this gap.

Warning — "E[X] = ∞" is a theorem, not a failure of definition

The definition does not "break" on Example 3; it computes, correctly, that the ticket prices of the approximating staircases grow without bound. Distinguish three verdicts the definition can return: integrable (both parts finite — the comfortable case), semi-infinite (E[X] = +∞ or −∞: well-defined as an extended real, unusable for linearity), and nonexistent (both infinite: e.g. the Cauchy variable, 5.5). Textbooks that quietly assume E|X| < ∞ are not being pedantic — every linearity manipulation in this section can fail for semi-infinite variables (try computing E[X − X + 1] naively on Example 3's variable and collect your ∞ − ∞). The course flags every such assumption from here on.

5.1.6First computations

Example 4 · The uniform, computed three ways

For the uniform on [0,1]: the banding sums give E[X] = 1/2 (each banding midpoint prices out to 1/2 by symmetry — Example 2(b) generalized). Second way, the layer-cake identity: for X ≥ 0, E[X] = ∫₀^∞ P(X > t) dt — for the uniform, ∫₀¹ (1−t) dt = 1/2. Third way, once 5.4 supplies change of variables, the familiar ∫₀¹ x dx. That three routes agree is not luck: it is Theorem 1 (well-posedness) plus the coming theorem that expectation depends only on the law (5.4). The layer-cake identity deserves a permanent place in a quant's toolkit: it converts expectations of positive variables into tail integrals — exactly the computation risk management does with VaR/ES, and exactly the shape that distinguishes heavy tails (5.5).

Example 5 · Same expectation, different variables: what the number misses

Three variables, all with expectation 0: (a) X ≡ 0; (b) a symmetric ±1 coin flip; (c) a symmetric ±1,000,000 coin flip. The expectation is blind to scale — to the spread, to the extremes, to everything a risk officer actually cares about. This is not a defect of our definition (the definition did exactly what an average does) but a boundary of the concept: the course's answer is to define more numbers (variance in 7.1's L² discussions, quantiles in 4.2) rather than to overload one. Any single-number summary of a distribution is a lossy compression; expectation is merely the first and the most-abused.

Expectation is the price of the ticket, computed as the limit of band prices — and the definition is honest precisely because it can return "infinite" or "does not exist".

Where we are, and what is next

We can price a ticket. We cannot yet do what the course most needs: take limits. Does X_n → X imply E[X_n] → E[X]? The answer is "sometimes, and the sometimes is the whole story": the next subchapter proves the three limit theorems — monotone convergence, Fatou's inequality, dominated convergence — and exhibits, for each, the example that shows what breaks when its hypothesis is dropped.