Expectation, integrability, and legitimate limit exchanges

For a nonnegative random variable, its expectation is defined by increasing simple approximations and may equal positive infinity. A real variable splits into its positive and negative parts. Its signed expectation is their difference only when the difference is not infinity minus infinity. If exactly one part has infinite integral, an extended expectation exists; if both do, the signed expectation is undefined. Declaring a distribution symmetric does not repair this undefined subtraction by cancellation.

Integrability means finite expectation of the absolute value. It guarantees that both parts have finite integrals and that the usual finite-valued linearity and triangle inequality apply. Square integrability is stronger on a probability space, by Cauchy–Schwarz, but integrability need not imply a finite variance. These distinctions are worth checking before using a variance-based estimate: a valid finite mean may coexist with a tail so heavy that the second moment diverges.

For a nonnegative variable, integrating its upper-tail probability over all nonnegative thresholds gives its expectation. This follows by writing its value as the length of an interval beneath it and applying Tonelli to the corresponding nonnegative indicator. The formula turns an integrability question into an area question. A slowly decaying tail can have infinite area even though the probability of exceeding any individual large threshold is small and tends to zero.

For example, a variable with density proportional to x to the power minus three on [1,infinity) has a finite first moment but an infinite second moment after normalization. Weighting its tail by another power of x changes which integral converges. The same analysis guides engineering models of large losses or service times, but a mathematical tail model must be stated explicitly. A finite data set cannot by itself establish that an unobserved tail has any particular asymptotic exponent.

Monotone convergence says that increasing nonnegative variables have expectations increasing to the expectation of their pointwise limit, allowing an infinite answer. Dominated convergence instead allows signed variables when they converge almost surely and are bounded in absolute value by one common integrable variable. That single dominator must work for the entire sequence. A bound depending on the index, or an almost-sure limit without tail control, does not satisfy the hypothesis.

The counterexample X_n=n times the indicator of U≤1/n for a uniform U makes the danger visible. Outside the null event U=0, the sequence eventually vanishes, yet every mean equals one. Increasing heights compensate for shrinking support. These variables are not dominated by an integrable common envelope: their supremum has a divergent tail integral. The lesson is not that expectations ignore typical values, but that a vanishing fraction of the probability space can retain all the integral mass.