Multidimensional Integration 1 | Lebesgue Measure and Lebesgue Integral [dark version]

Q: How does the construction of Lebesgue measure move from interval lengths to a measure on many sets?

It begins with a pre-measure on intervals: for $[a,b]$ with $b\ge a$, set $\mu([a,b])=b-a$. That rule only works on intervals, so an outer measure is formed and then extended using the Carathéodory extension theorem (named “K theodoris extension theorem” in the transcript). The result is a genuine measure $\Lambda$ defined on a $\sigma$-algebra of Lebesgue measurable sets $\mathcal{L}(\mathbb{R})$, not on all subsets of $\mathbb{R}$.

Q: What does $\sigma$-additivity mean for Lebesgue measure, and why does it require disjoint sets?

For pairwise disjoint measurable sets $A_j$, $\Lambda\left(\bigcup_{j=1}^\infty A_j\right)=\sum_{j=1}^\infty \Lambda(A_j)$. Disjointness ensures the union doesn’t double-count points, so the total measure is the sum of the parts. The countable nature of the union is why a $\sigma$-algebra is needed.

Q: Why are null sets important, and what special property do they have under Lebesgue measure?

A null set is a measurable set $A$ with $\Lambda(A)=0$. The transcript stresses a key property: if $A$ is null, then every subset $B\subseteq A$ is also null (and measurable). This goes beyond what one might expect from length comparisons alone and becomes useful when defining integrals and handling “negligible” sets.

Q: How is the Lebesgue integral defined using simple functions, and what does the supremum represent?

For nonnegative $f$, simple functions take finitely many values and lie pointwise below $f$. Each simple function partitions $\mathbb{R}$ into level sets (preimages of each value), and the integral of the simple function is computed by summing “value × measure of the level set,” giving an area-like approximation. The Lebesgue integral is the supremum over all such lower approximations, so it captures the largest area consistent with approximating from below.

TL;DR

Lebesgue measure $Λ$ is constructed by extending interval length $b - a$ from a pre-measure to a full measure on the $σ$ -algebra of Lebesgue measurable sets $L (R)$ .

Briefing Cornell Notes

Briefing

Multidimensional integration in higher-dimensional spaces is built on a single foundation: the Lebesgue measure and the Lebesgue integral. The core payoff is that this framework turns “length” and “area under a curve” into measure-theoretic objects that behave predictably under limits, making it possible to generalize integration rules and compute integrals in ways that Riemann integration can’t.

The starting point is the one-dimensional Lebesgue measure on $R$ . Intervals $[a, b]$ are assigned length $μ ([a, b]) = b - a$ (for $b \geq a$ ), which initially defines only a “pre-measure” on intervals. A key step then extends this interval-based rule to a full measure defined on a $σ$ -algebra of sets. The construction uses an outer measure and the Carathéodory extension theorem (referred to as “K theodoris extension theorem”), producing a measure $Λ$ on the collection of Lebesgue measurable sets $L (R)$ . Not every subset of $R$ is measurable, but the measurable sets are broad enough for essentially all practical applications.

Lebesgue measure $Λ$ satisfies the defining measure axioms: $Λ (\emptyset) = 0$ and $σ$ -additivity, meaning that for pairwise disjoint measurable sets $A_{j}$ , the measure of their union equals $\sum_{j = 1}^{\infty} Λ (A_{j})$ . Beyond these basics, Lebesgue measurable sets form a $σ$ -algebra that contains the Borel $σ$ -algebra, so all open and closed sets are measurable. Sets of measure zero are called null sets, and an important subtlety follows: if $A$ is null, then every subset of $A$ is also null (and measurable). This “subset of a null set stays measurable” property becomes crucial later when defining and manipulating integrals.

Two additional structural properties are emphasized. First, normalization: an interval of length one has measure one. Second, translation invariance: shifting a measurable set by any real number $x$ does not change its Lebesgue measure. This invariance is highlighted because it will persist in higher dimensions.

Once the measure is in place, the Lebesgue integral is defined with respect to $Λ$ . The integral is written in several equivalent notations, such as $\int_{A} f d Λ$ or $\int_{A} f (x) d Λ (x)$ , and in this course the measure symbol is often suppressed, leaving $d x$ as shorthand for integration with respect to Lebesgue measure. The construction uses approximation by simple functions (step functions in the Lebesgue sense): for nonnegative $f$ , simple functions take only finitely many values and sit below $f$ . Their integrals correspond to “areas” computed by summing value times measure of the preimage sets. The Lebesgue integral is then the supremum of these lower approximations. This approach matches the Riemann integral when Riemann integration works, but extends to cases where Riemann fails—one reason the Lebesgue framework is adopted for the move to $R^{n}$ , where the next step is defining $n$ -dimensional Lebesgue measure and integral, along with tools like change of variables and Fubini–Tonelli for iterated integration.

Cornell Notes

The course builds integration in $R^{n}$ on the Lebesgue measure $Λ$ and Lebesgue integral. It starts in one dimension by assigning interval length $b - a$ to $[a, b]$ , then uses an extension theorem to turn that interval rule into a full measure on the $σ$ -algebra of Lebesgue measurable sets $L (R)$ . Lebesgue measure is $σ$ -additive, contains all Borel sets (so open/closed sets are measurable), and has null sets with the property that every subset of a null set is null. The Lebesgue integral is defined via approximation from below by simple functions, taking the supremum of their “area” calculations. This matches the Riemann integral when Riemann applies, but succeeds in broader cases and sets up easier generalization to higher dimensions.

How does the construction of Lebesgue measure move from interval lengths to a measure on many sets?

It begins with a pre-measure on intervals: for

[a, b]

with

b \geq a

, set

μ ([a, b]) = b - a

. That rule only works on intervals, so an outer measure is formed and then extended using the Carathéodory extension theorem (named “K theodoris extension theorem” in the transcript). The result is a genuine measure

Λ

defined on a

σ

-algebra of Lebesgue measurable sets

L (R)

, not on all subsets of

R

What does $σ$ -additivity mean for Lebesgue measure, and why does it require disjoint sets?

For pairwise disjoint measurable sets

A_{j}

Λ (⋃_{j = 1}^{\infty} A_{j}) = \sum_{j = 1}^{\infty} Λ (A_{j})

. Disjointness ensures the union doesn’t double-count points, so the total measure is the sum of the parts. The countable nature of the union is why a

σ

-algebra is needed.

Why are null sets important, and what special property do they have under Lebesgue measure?

A null set is a measurable set

A

with

Λ (A) = 0

. The transcript stresses a key property: if

A

is null, then every subset

B \subseteq A

is also null (and measurable). This goes beyond what one might expect from length comparisons alone and becomes useful when defining integrals and handling “negligible” sets.

What two structural properties of Lebesgue measure are highlighted besides the axioms?

Normalization and translation invariance. Normalization means intervals behave like expected lengths—for example, the unit interval has measure one. Translation invariance means that for any measurable set

A

and any real shift

x

, the shifted set

A + x

has the same Lebesgue measure as

A

How is the Lebesgue integral defined using simple functions, and what does the supremum represent?

For nonnegative

f

, simple functions take finitely many values and lie pointwise below

f

. Each simple function partitions

R

into level sets (preimages of each value), and the integral of the simple function is computed by summing “value × measure of the level set,” giving an area-like approximation. The Lebesgue integral is the supremum over all such lower approximations, so it captures the largest area consistent with approximating from below.

Review Questions

What role does the $σ$ -algebra $L (R)$ play in making Lebesgue measure well-defined?
Explain how Lebesgue integration approximates a nonnegative function from below and why taking a supremum matters.
List at least three properties of Lebesgue measure mentioned in the transcript (e.g., $σ$ -additivity, null sets, translation invariance) and state what each implies.

Key Points

1
Lebesgue measure $Λ$ is constructed by extending interval length $b - a$ from a pre-measure to a full measure on the $σ$ -algebra of Lebesgue measurable sets $L (R)$ .
2
Lebesgue measure satisfies $Λ (\emptyset) = 0$ and $σ$ -additivity for countable unions of pairwise disjoint measurable sets.
3
All Borel sets are Lebesgue measurable, so open and closed sets are measurable under $Λ$ .
4
Null sets (sets with $Λ (A) = 0$ ) have the strong property that every subset of a null set is also null and measurable.
5
Lebesgue measure is translation invariant: shifting a measurable set by any real number does not change its measure.
6
The Lebesgue integral is defined for nonnegative functions via approximation from below by simple functions and taking the supremum of their integrals.
7
The Lebesgue integral matches the Riemann integral when Riemann integration applies, but extends to cases where Riemann fails, enabling a smoother transition to $R^{n}$ .

Highlights

Lebesgue measure turns “length of intervals” into a full measure on a σ-algebra, enabling σ-additivity across countable unions.

The transcript emphasizes a powerful null-set fact: every subset of a measure-zero set is measurable and also has measure zero.

Lebesgue integration is built from below using simple functions, with the integral defined as the supremum of these area approximations.

Translation invariance and normalization are presented as core structural properties that will carry into higher dimensions.

Topics

Lebesgue Measure
Lebesgue Integral
Measure Extension
Simple Functions
Fubini Tonelli