the Lebesgue integral as expectation
/ luh-BEG /
The expectation E[X] is supposed to be the average value of a random variable, but for continuous variables, weird mixtures, or limits of variables, the elementary formulas (a weighted sum for discrete, an ordinary integral for continuous) are two different recipes that do not always apply. The Lebesgue integral is a single, more powerful way of averaging that handles all of these at once. Expectation, rigorously, is defined as the Lebesgue integral of X against the probability measure: E[X] = integral over Omega of X dP.
The idea differs from the calculus integral in a clever way. The Riemann integral slices the horizontal axis into thin vertical strips and sums their areas — it partitions the domain. The Lebesgue integral instead slices the vertical axis: it partitions the range of values, asks how much probability is attached to each thin band of values, multiplies value by that probability, and sums. Built up in stages, it first defines the integral of a simple function (one taking finitely many values c1, c2, ...) as the obvious sum c1 P(X = c1) + c2 P(X = c2) + ..., then defines the integral of any non-negative measurable X as the supremum of the integrals of simple functions below it, and finally handles signed X by splitting into positive and negative parts. This horizontal-slicing view is exactly why E[X] = sum of x times P(X = x) in the discrete case and E[X] = integral of x f(x) dx in the continuous case both fall out as special instances of one definition.
The payoff is robustness. The Lebesgue integral converges for far more functions than the Riemann integral, and — crucially — it comes with powerful limit theorems (monotone convergence, Fatou, dominated convergence) that say precisely when you may swap a limit with an expectation. That single, uniform definition is what lets probability prove the law of large numbers, define expectations for any distribution, and pass limits through integrals safely, where ad hoc calculus would stall.
For a discrete X taking value 1 with probability 1/3 and value 4 with probability 2/3, the Lebesgue integral of the simple function gives E[X] = 1 times 1/3 + 4 times 2/3 = 3. For a continuous X with density f, the same definition collapses to the familiar E[X] = integral of x f(x) dx — one rule, both cases.
Slice by value, not by domain: one definition of averaging that subsumes both the discrete sum and the continuous integral.
E[X] exists in the Lebesgue sense only when E[|X|] is finite; a heavy-tailed variable like the Cauchy has no expectation at all, because the positive and negative parts both integrate to infinity.