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A set is Lebesgue measurable if it is part of the sigma-algebra on which the Lebesgue measure is defined. Measurability ensures that the set can be appropriately assigned a measure.

The outer measure of a set is the infimum of the measures of open covers of the set. It represents an 'external' approach to measuring the size of the set and is not necessarily additive.

In measure theory, a measure is a function that assigns a non-negative real number or $\infty$ to subsets of a set, provided they are in a sigma-algebra, in a way that satisfies certain properties like countable additivity.

Fatou's Lemma states that for any sequence of non-negative measurable functions $\{f_n\}$, the integral of the limit inferior of the functions is less than or equal to the limit inferior of the integrals, i.e., $\int \liminf f_n d\mu \leq \liminf \int f_n d\mu$.

The Lebesgue integral extends the idea of integration to a wider class of functions and is integral to the study of modern analysis. It differs from the Riemann integral by its way of approaching the construction of the integral, focusing on the measure of the range rather than the domain.

A null function is a function that is zero almost everywhere in the domain. In measure theory, two functions are considered equivalent if they differ only on a null set.

The Borel sigma-algebra on a topological space is the sigma-algebra generated by the open sets (or equivalently, the closed sets). For the real numbers, this is the smallest sigma-algebra containing all intervals.

A measure space is a triple $(X, \mathcal{F}, \mu)$ consisting of a set $X$, a sigma-algebra $\mathcal{F}$ of subsets of $X$, and a measure $\mu$ on $\mathcal{F}$. It's the fundamental structure in measure theory.

Countable additivity is a property of a measure which states that the measure of a countable union of disjoint sets is equal to the sum of the measures of each set, i.e., $\mu(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} \mu(A_i)$ for disjoint $A_i$'s.

A null set is a set of measure zero within a given measure space. It's important in the sense that modifications on a null set do not affect integrals or probability outcomes.

A measure space is complete if every subset of every null set is also measurable (and hence is a null set). Such additional subsets do not need to be explicitly included in the sigma-algebra.

Monotonicity is the property that if $A \subseteq B$, then $\mu(A) \leq \mu(B)$ for any measure $\mu$. It reflects the intuitive notion that the measure should not decrease if the set becomes larger.

Carathéodory's Criterion is a theorem that provides a condition for when a set is measurable with respect to a measure defined by an outer measure: a set E is measurable if for every set A, $m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)$, where $m^*$ denotes the outer measure.

Lebesgue's Differentiation Theorem asserts that if a function is locally integrable on $\mathbb{R}^n$, then for almost every point, the function value at that point equals the limit of the averages taken over hyperspheres or hypercubes that shrink to that point.

Fubini's Theorem gives conditions under which a double integral of a function over a product measure space can be computed as an iterated integral, allowing the computation of the integral to be broken down into the computation of two separate integrals.

This theorem states that if a sequence of measurable functions converges pointwise almost everywhere to a function f and is dominated by some integrable function, then the limit function is integrable and the integral of the limit function equals the limit of the integrals of the functions in the sequence.

Sub-additivity is the property of a measure which asserts that the measure of a union of sets is at most the sum of the measures of the individual sets, i.e., $\mu(\bigcup_{i=1}^{\infty} A_i) \leq \sum_{i=1}^{\infty} \mu(A_i)$, without the requirement that the $A_i$ be disjoint.

The Lebesgue measure is the standard way of assigning a measure to subsets of $\mathbb{R}^n$, generalizing the notion of length, area, and volume. It is translation invariant and is defined on the sigma-algebra of Lebesgue measurable sets.

A sigma-algebra is a collection of subsets of a given set X that is closed under complementation and countable unions. It forms the technical foundation for a measure space.