cubeintegrable

Cubeintegrable is a term used to describe a function or random variable whose cube is integrable. In measure theory and functional analysis, a measurable function f on a measure space (X, Σ, μ) is cube-integrable if the integral of its absolute value raised to the third power is finite, that is, ∫_X |f(x)|^3 dμ(x) < ∞. Functions with this property belong to the Lebesgue space L^3(μ). In probability theory, cube-integrable corresponds to having a finite third moment: E(|X|^3) < ∞ for a random variable X.

Basic properties and examples

If X has finite measure, then every cube-integrable function is also integrable to lower powers; in particular, L^3(μ) is contained in L^2(μ) and L^1(μ) on spaces with finite measure. Conversely, a function may be cube-integrable without being square-integrable on spaces of infinite measure. Bounded functions are cube-integrable on spaces of finite measure, since ∫ |f|^3 ≤ sup|f|^3 · μ(X). On the unit interval with Lebesgue measure, for example, the function f(x) = x is cube-integrable because ∫_0^1 |x|^3 dx = 1/4.

Relation to other concepts

Cube-integrability is the case p = 3 in the family of L^p spaces. It interacts with Hölder’s inequality and other standard tools from real analysis. Finite third moments are relevant in probability for skewness measures and in limit theorems, while in analysis L^3 functions arise in interpolation, PDEs, and harmonic analysis.

In summary, cubeintegrable typically denotes belonging to L^3, i.e., having finite integral of the cube of the absolute value, with the associated third-moment interpretation in probabilistic contexts.