The Central Idea
Measure theory separates integration into two structural questions:
- Which sets can be assigned a meaningful size?
- Which functions interact correctly with those measurable sets?
The central progression is
\[ \boxed{ \text{measurable sets} \longrightarrow \text{measures} \longrightarrow \text{measurable functions} \longrightarrow \text{Lebesgue integration} }. \]Why Measure Theory Is Needed
The Riemann integral works extremely well for continuous functions and many mildly discontinuous functions, but it can behave poorly under limits.
Modern analysis needs a framework that can:
- assign size to complicated subsets of \(\mathbb{R}\);
- distinguish negligible sets from substantial ones;
- handle functions with many discontinuities;
- support strong convergence theorems;
- build function spaces such as \(L^1\) and \(L^2\);
- make integration stable under useful limiting processes.
From Length to Measure
For an interval,
\[ [a,b], \]the natural notion of length is
\[ b-a. \]Measure theory asks whether this idea can be extended consistently to much more complicated sets while preserving familiar rules such as nonnegativity, additivity, and invariance under suitable decompositions.
The Underlying Set
Measure theory begins with a set \(X\), often called the underlying space. Examples include
\[ X=\mathbb{R}, \qquad X=\mathbb{R}^n, \qquad X=\mathbb{N}. \]We then select a collection of subsets of \(X\) that will be declared measurable.
What Is a Sigma-Algebra?
A collection \(\mathcal{M}\) of subsets of \(X\) is a sigma-algebra if:
- \(X\in\mathcal{M}\);
- if \(E\in\mathcal{M}\), then \(X\setminus E\in\mathcal{M}\);
- if \(E_1,E_2,\ldots\in\mathcal{M}\), then \[ \bigcup_{n=1}^{\infty}E_n\in\mathcal{M}. \]
The pair
\[ (X,\mathcal{M}) \]is called a measurable space.
Why Countable Unions Matter
Limits naturally generate countable operations. A sequence of sets may be combined through
\[ \bigcup_{n=1}^{\infty} E_n \]or
\[ \bigcap_{n=1}^{\infty} E_n. \]Closure under countable unions ensures that measurable structure survives the kinds of constructions that arise in sequences, limits, and integration.
Closure Under Countable Intersections
A sigma-algebra is automatically closed under countable intersections. By De Morgan's law,
\[ \bigcap_{n=1}^{\infty}E_n = X\setminus \bigcup_{n=1}^{\infty} (X\setminus E_n). \]Since complements and countable unions remain measurable, so do countable intersections.
The Smallest Sigma-Algebra
The collection
\[ \{\varnothing,X\} \]is always a sigma-algebra on \(X\).
It contains only the empty set and the whole space, so it is called the trivial sigma-algebra.
The Largest Sigma-Algebra
The power set
\[ \mathcal{P}(X) \]is also a sigma-algebra. It contains every subset of \(X\).
In finite spaces, using the full power set is often natural. On \(\mathbb{R}\), however, the sigma-algebras used in analysis are more selective.
Sigma-Algebras Generated by Collections
Given a collection \(\mathcal{C}\) of subsets of \(X\), there is a smallest sigma-algebra containing \(\mathcal{C}\). It is denoted
\[ \sigma(\mathcal{C}). \]Conceptually, we begin with the sets in \(\mathcal{C}\) and close under all operations required by the sigma-algebra axioms.
The Borel Sigma-Algebra
On \(\mathbb{R}\), the Borel sigma-algebra is the sigma-algebra generated by the open sets:
\[ \boxed{ \mathcal{B}(\mathbb{R}) = \sigma(\text{open subsets of }\mathbb{R}) }. \]Borel sets include all open sets, closed sets, intervals, countable unions of such sets, countable intersections of such sets, and all sets obtained from them through countably many sigma-algebra operations.
Equivalent Generators of the Borel Sigma-Algebra
The same Borel sigma-algebra can be generated by many simpler families, such as
\[ \{(a,b):aThis flexibility is useful because measurable-function proofs often reduce to checking inverse images of half-lines.
What Is a Measure?
Let \((X,\mathcal{M})\) be a measurable space. A measure is a function
\[ \mu:\mathcal{M}\to[0,\infty] \]satisfying:
- \[ \mu(\varnothing)=0; \]
- if \(E_1,E_2,\ldots\) are pairwise disjoint measurable sets, then \[ \boxed{ \mu\left(\bigcup_{n=1}^{\infty}E_n\right) = \sum_{n=1}^{\infty}\mu(E_n) }. \]
The second property is called countable additivity.
A Measure Space
A triple
\[ (X,\mathcal{M},\mu) \]is called a measure space.
The set \(X\) provides the objects, the sigma-algebra \(\mathcal{M}\) determines which subsets are measurable, and the measure \(\mu\) assigns a size to each measurable set.
Finite and Sigma-Finite Measures
A measure is finite if
\[ \mu(X)<\infty. \]It is sigma-finite if
\[ X=\bigcup_{n=1}^{\infty}E_n \]for measurable sets \(E_n\) satisfying
\[ \mu(E_n)<\infty. \]Lebesgue measure on \(\mathbb{R}\) is not finite, but it is sigma-finite because
\[ \mathbb{R} = \bigcup_{n=1}^{\infty}[-n,n]. \]The Counting Measure
On any set \(X\), define
\[ \mu(E) = \#E \]when \(E\) is finite, and
\[ \mu(E)=\infty \]when \(E\) is infinite.
This is called counting measure. It translates the concept of cardinality into measure-theoretic language.
The Dirac Measure
Fix a point \(a\in X\). The Dirac measure at \(a\) is defined by
\[ \delta_a(E) = \begin{cases} 1, & a\in E,\\ 0, & a\notin E. \end{cases} \]All of the measure is concentrated at one point.
Lebesgue Measure
Lebesgue measure on \(\mathbb{R}\), denoted here by \(m\), extends ordinary interval length:
\[ m([a,b])=b-a. \]The same value is obtained for
\[ (a,b), \qquad [a,b), \qquad (a,b]. \]Individual endpoints have measure zero.
Singletons Have Lebesgue Measure Zero
For any \(a\in\mathbb{R}\),
\[ \boxed{m(\{a\})=0}. \]One way to understand this is that \(\{a\}\) can be enclosed in intervals of arbitrarily small length.
Countable Sets Have Measure Zero
If
\[ E=\{x_1,x_2,\ldots\} \]is countable, then by countable subadditivity,
\[ m(E) \le \sum_{n=1}^{\infty}m(\{x_n\}) = 0. \]Therefore,
\[ \boxed{m(E)=0}. \]The Rational Numbers Have Measure Zero
Since \(\mathbb{Q}\) is countable,
\[ \boxed{m(\mathbb{Q})=0}. \]This does not mean the rational numbers are absent or topologically small in every sense. In fact, \(\mathbb{Q}\) is dense in \(\mathbb{R}\). Measure and density describe different kinds of structure.
Null Sets
A measurable set \(N\) is a null set if
\[ \mu(N)=0. \]Null sets are negligible from the viewpoint of the measure. Many central statements in analysis are permitted to fail on a null set.
Almost Everywhere
A property is said to hold almost everywhere, often written a.e., if the set where it fails has measure zero.
For example,
\[ f=g \quad\text{a.e.} \]means
\[ \mu(\{x:f(x)\ne g(x)\})=0. \]Almost Everywhere Does Not Mean Everywhere
Suppose
\[ f(x)=0 \]for all \(x\), and define
\[ g(x)= \begin{cases} 1, & x=0,\\ 0, & x\ne0. \end{cases} \]Then \(f\ne g\) as pointwise functions, but
\[ \boxed{f=g\text{ a.e.}} \]with respect to Lebesgue measure.
Monotonicity of Measures
If
\[ A\subseteq B, \]then
\[ \boxed{\mu(A)\le\mu(B)}. \]This follows by writing
\[ B=A\cup(B\setminus A) \]as a disjoint union.
Countable Subadditivity
For measurable sets \(E_1,E_2,\ldots\),
\[ \boxed{ \mu\left( \bigcup_{n=1}^{\infty}E_n \right) \le \sum_{n=1}^{\infty}\mu(E_n) }. \]Equality is guaranteed when the sets are pairwise disjoint.
Continuity from Below
Suppose
\[ E_1\subseteq E_2\subseteq E_3\subseteq\cdots. \]Then
\[ \boxed{ \mu\left( \bigcup_{n=1}^{\infty}E_n \right) = \lim_{n\to\infty}\mu(E_n) }. \]This property is called continuity from below.
Continuity from Above
Suppose
\[ E_1\supseteq E_2\supseteq E_3\supseteq\cdots \]and assume
\[ \mu(E_1)<\infty. \]Then
\[ \boxed{ \mu\left( \bigcap_{n=1}^{\infty}E_n \right) = \lim_{n\to\infty}\mu(E_n) }. \]The finiteness assumption on the first set is important.
Why the Finiteness Hypothesis Matters
Let
\[ E_n=[n,\infty) \]in \(\mathbb{R}\) with Lebesgue measure. Then
\[ E_1\supseteq E_2\supseteq\cdots \]and
\[ \bigcap_{n=1}^{\infty}E_n=\varnothing. \]But every \(E_n\) has infinite measure. Thus
\[ \lim_{n\to\infty}m(E_n)=\infty \]while
\[ m\left(\bigcap E_n\right)=0. \]This shows why continuity from above requires a finite-measure hypothesis.
Outer Measure
One route to Lebesgue measure begins by assigning an outer measure to every subset \(E\subseteq\mathbb{R}\):
\[ m^*(E) = \inf \left\{ \sum_{n=1}^{\infty}|I_n| : E\subseteq\bigcup_{n=1}^{\infty}I_n \right\}, \]where the \(I_n\) are intervals and \(|I_n|\) denotes their lengths.
Outer measure asks how efficiently a set can be covered by intervals.
Caratheodory Measurability
A set \(E\subseteq\mathbb{R}\) is Lebesgue measurable if for every set \(A\subseteq\mathbb{R}\),
\[ \boxed{ m^*(A) = m^*(A\cap E) + m^*(A\setminus E) }. \]The condition says that \(E\) cuts every test set \(A\) cleanly from the perspective of outer measure.
Borel Measurable versus Lebesgue Measurable
Every Borel set is Lebesgue measurable, but the collection of Lebesgue measurable sets is larger.
Lebesgue measurability includes all Borel sets together with subsets of Lebesgue null sets and sets obtained through the sigma-algebra operations.
This enlargement is called a completion of the Borel measure structure.
Complete Measure Spaces
A measure space is complete if every subset of every measurable null set is itself measurable.
Lebesgue measure on the Lebesgue measurable sets is complete.
Borel measure on the Borel sigma-algebra is not complete because a subset of a Borel null set need not itself be Borel.
What Is a Measurable Function?
Let \((X,\mathcal{M})\) be a measurable space. A function
\[ f:X\to\mathbb{R} \]is measurable if for every Borel set \(B\subseteq\mathbb{R}\),
\[ \boxed{ f^{-1}(B)\in\mathcal{M} }. \]In other words, measurable functions pull measurable target sets back to measurable domain sets.
Testing Measurability with Half-Lines
It is enough to verify that
\[ \{x:f(x)Equivalent tests use sets such as
\[ \{x:f(x)\le a\}, \qquad \{x:f(x)>a\}, \qquad \{x:f(x)\ge a\}. \]These tests work because half-lines generate the Borel sigma-algebra.
Continuous Functions Are Borel Measurable
If
\[ f:\mathbb{R}\to\mathbb{R} \]is continuous, then the inverse image of every open set is open.
Since every open set is Borel,
\[ \boxed{ \text{continuous} \Longrightarrow \text{Borel measurable} }. \]Measurability is therefore weaker than continuity.
Indicator Functions
For a set \(E\subseteq X\), define its indicator function by
\[ \mathbf{1}_E(x) = \begin{cases} 1, & x\in E,\\ 0, & x\notin E. \end{cases} \]The indicator function \(\mathbf{1}_E\) is measurable exactly when \(E\) is measurable.
Indicator functions provide the basic link between sets and functions.
Simple Functions
A simple function is a measurable function taking only finitely many values.
It can be written in the form
\[ \boxed{ \phi = \sum_{k=1}^{n} a_k\mathbf{1}_{E_k} } \]where the \(E_k\) are measurable sets.
Simple functions play the role of building blocks for the Lebesgue integral.
Example of a Simple Function
Define
\[ \phi(x) = \begin{cases} 2, & x\in[0,1),\\ 5, & x\in[1,3],\\ 0, & \text{otherwise}. \end{cases} \]Then
\[ \phi = 2\mathbf{1}_{[0,1)} + 5\mathbf{1}_{[1,3]}. \]Since the intervals are measurable, \(\phi\) is measurable.
Algebra of Measurable Functions
If \(f\) and \(g\) are measurable real-valued functions, then the following are measurable:
\[ f+g, \qquad f-g, \qquad fg, \]and, wherever the quotient is properly defined,
\[ \frac{f}{g}. \]Also measurable are
\[ |f|, \qquad \max(f,g), \qquad \min(f,g). \]Limits of Measurable Functions
Suppose each \(f_n\) is measurable. Then the pointwise functions
\[ \sup_n f_n, \qquad \inf_n f_n, \qquad \limsup_{n\to\infty}f_n, \qquad \liminf_{n\to\infty}f_n \]are measurable, with the usual extended-real interpretation when needed.
In particular, if
\[ f_n(x)\to f(x) \]pointwise, then \(f\) is measurable.
The Limsup and Liminf of Functions
The pointwise upper limit is
\[ \limsup_{n\to\infty} f_n = \inf_{N\ge1}\sup_{n\ge N}f_n, \]and the lower limit is
\[ \liminf_{n\to\infty} f_n = \sup_{N\ge1}\inf_{n\ge N}f_n. \]Pointwise convergence occurs exactly where these two values agree and are finite.
Positive and Negative Parts
For a real-valued function \(f\), define
\[ f^+(x)=\max(f(x),0) \]and
\[ f^-(x)=\max(-f(x),0). \]Then
\[ \boxed{ f=f^+-f^- } \]and
\[ \boxed{ |f|=f^++f^-. } \]This decomposition becomes fundamental in defining the Lebesgue integral for signed functions.
Approximation by Simple Functions
Every nonnegative measurable function can be approximated from below by an increasing sequence of nonnegative simple functions.
That is, if
\[ f:X\to[0,\infty] \]is measurable, then there exist simple functions \(\phi_n\) such that
\[ 0\le\phi_1\le\phi_2\le\cdots\le f \]and
\[ \boxed{ \phi_n(x)\to f(x) } \]pointwise.
Why Simple Approximation Matters
The Lebesgue integral is first defined for simple functions. The integral of a general nonnegative measurable function is then built by taking the supremum of the integrals of simple functions lying below it.
This is the measure-theoretic analogue of approximating complicated objects by simpler ones while preserving monotone control.
Almost-Everywhere Convergence
A sequence \(f_n\) converges to \(f\) almost everywhere if
\[ f_n(x)\to f(x) \]for every \(x\) outside a null set.
Symbolically,
\[ \boxed{ f_n\to f \quad\text{a.e.} }. \]Pointwise versus Almost-Everywhere Convergence
Pointwise convergence requires convergence at every point in the stated domain.
Almost-everywhere convergence permits failure on a measure-zero exceptional set.
Thus
\[ \text{pointwise convergence} \Longrightarrow \text{a.e. convergence}, \]but the converse need not hold.
Convergence in Measure
On a measure space, \(f_n\) converges to \(f\) in measure if for every \(\varepsilon>0\),
\[ \boxed{ \mu\bigl( \{x:|f_n(x)-f(x)|>\varepsilon\} \bigr) \to0 }. \]Instead of requiring pointwise closeness, convergence in measure requires the set of points with substantial error to become small in measure.
Almost-Everywhere Convergence and Convergence in Measure
On a finite measure space,
\[ f_n\to f \quad\text{a.e.} \]implies
\[ f_n\to f \quad\text{in measure}. \]The converse is not generally true for the full sequence, although convergence in measure does guarantee the existence of a subsequence that converges almost everywhere under standard measure-space assumptions.
Equality Almost Everywhere
If
\[ f=g \quad\text{a.e.}, \]then many measure-theoretic constructions treat \(f\) and \(g\) as equivalent.
This becomes essential in spaces such as \(L^p\), where functions that differ only on null sets represent the same element.
Measurable Sets from Measurable Functions
If \(f\) is measurable, then sets defined by inequalities involving \(f\) are measurable. Examples include
\[ \{x:f(x)>0\}, \] \[ \{x:|f(x)|\le M\}, \]and
\[ \{x:aLevel Sets
For a function \(f\), a typical level or superlevel set is
\[ \{x:f(x)>\alpha\}. \]Measurability ensures that these sets belong to the sigma-algebra, so their sizes can be measured.
Level sets later become central in inequalities, probability, and integration theory.
Measurability Is Not the Same as Continuity
Every continuous real-valued function is Borel measurable, but measurable functions may be highly discontinuous.
For example, the indicator function
\[ \mathbf{1}_{\mathbb{Q}} \]is Lebesgue measurable, even though it is discontinuous at every real number.
Measurability Is Not the Same as Integrability
A measurable function need not have a finite integral.
For example,
\[ f(x)=\frac1x \]on \((0,1]\) is measurable, but its integral over \((0,1]\) is infinite in the Lebesgue sense.
Measurability makes integration possible to define. It does not guarantee finiteness.
Common Error: Treating Every Subset as Measurable
In many elementary examples, every subset is measurable. On \(\mathbb{R}\), however, not every subset is Lebesgue measurable.
Measure theory therefore always keeps track of the sigma-algebra as part of the space.
Common Error: Confusing Measure Zero with the Empty Set
A null set can contain infinitely many points.
For example,
\[ m(\mathbb{Q})=0, \]yet \(\mathbb{Q}\) is infinite and dense in \(\mathbb{R}\).
Common Error: Confusing Dense with Large Measure
Density is topological. Measure is quantitative.
The rationals are dense but have measure zero. The irrationals are also dense and have full Lebesgue measure in every interval.
These concepts answer different questions.
Common Error: Forgetting Countable Additivity Requires Disjointness
The identity
\[ \mu\left( \bigcup_n E_n \right) = \sum_n\mu(E_n) \]is guaranteed only when the \(E_n\) are pairwise disjoint.
Without disjointness, the correct general statement is the inequality
\[ \mu\left( \bigcup_n E_n \right) \le \sum_n\mu(E_n). \]Common Error: Assuming Measurable Means Continuous
Continuity implies measurability, but not conversely.
Indicator functions of measurable sets often provide simple counterexamples.
Common Error: Ignoring the Exceptional Set in an A.E. Statement
Saying
\[ f=g \quad\text{a.e.} \]does not mean \(f(x)=g(x)\) for every \(x\).
A rigorous statement should identify that the failure set has measure zero.
A Reliable Procedure for Checking a Sigma-Algebra
- Verify that \(X\) belongs to the collection.
- Check closure under complements relative to \(X\).
- Check closure under countable unions.
- Use De Morgan's laws to infer closure under countable intersections.
- Test edge cases such as \(\varnothing\) and repeated sets.
A Reliable Procedure for Checking a Measure
- Identify the measurable space \((X,\mathcal{M})\).
- Verify that the proposed values lie in \([0,\infty]\).
- Check that \(\mu(\varnothing)=0\).
- Take an arbitrary sequence of pairwise disjoint measurable sets.
- Verify countable additivity on their union.
- Derive useful consequences such as monotonicity and subadditivity only after the axioms are established.
A Reliable Procedure for Checking Measurability of a Function
- Identify the domain sigma-algebra.
- Choose a generating family in the target, usually half-lines.
- Compute an inverse image such as \(\{x:f(x)
- Show that this inverse image belongs to the domain sigma-algebra for every \(a\).
- Use closure rules and known measurable-function operations when possible.
- Do not substitute continuity for measurability unless continuity is actually known.
A Reliable Procedure for Proving a Set Has Measure Zero
- Cover the set by measurable sets whose total measure is easy to estimate.
- Arrange the covering so the total measure is less than an arbitrary \(\varepsilon>0\).
- Apply countable subadditivity.
- Conclude that the measure is at most every positive \(\varepsilon\).
- Therefore conclude that the measure is zero.
Example: A Countable Null Set
Let
\[ E=\{x_1,x_2,\ldots\}. \]Given \(\varepsilon>0\), cover \(x_n\) by an interval \(I_n\) of length
\[ \frac{\varepsilon}{2^{n+1}}. \]Then
\[ E\subseteq\bigcup_{n=1}^{\infty}I_n \]and
\[ \sum_{n=1}^{\infty}|I_n| = \frac{\varepsilon}{2}. \]Since \(\varepsilon\) was arbitrary, the outer measure of \(E\) is zero.
Example: Measurability of \(f(x)=x^2\)
The function
\[ f(x)=x^2 \]is continuous on \(\mathbb{R}\). Therefore the inverse image of every open set is open.
Since open sets are Borel measurable,
\[ \boxed{x^2\text{ is Borel measurable}}. \]Example: Measurability of an Indicator Function
Let
\[ f=\mathbf{1}_{[0,1]}. \]Since \([0,1]\) is Borel measurable, the inverse image of each Borel set under \(f\) can only be one of a few sets:
\[ \varnothing, \qquad [0,1], \qquad \mathbb{R}\setminus[0,1], \qquad \mathbb{R}. \]All are Borel measurable. Hence \(f\) is measurable.
Example: Continuity from Below
Let
\[ E_n=[0,1-\tfrac1n] \qquad (n\ge2). \]Then
\[ E_2\subseteq E_3\subseteq\cdots \]and
\[ \bigcup_{n=2}^{\infty}E_n=[0,1). \]Lebesgue measure gives
\[ m(E_n)=1-\frac1n. \]Therefore,
\[ m([0,1)) = \lim_{n\to\infty} \left(1-\frac1n\right) = \boxed{1}. \]Practice 1: Sigma-Algebra
Let
\[ X=\{1,2,3\}. \]Determine whether
\[ \mathcal{M} = \{\varnothing,\{1\},\{2,3\},X\} \]is a sigma-algebra.
Practice 2: Counting Measure
Using counting measure on \(\mathbb{N}\), find
\[ \mu(\{2,4,6,8\}) \]and
\[ \mu(\{2,4,6,\ldots\}). \]Practice 3: Dirac Measure
Let \(\delta_0\) be the Dirac measure at \(0\). Compute
\[ \delta_0([-1,1]), \qquad \delta_0((1,3)). \]Practice 4: Measure Zero
Explain why every finite subset of \(\mathbb{R}\) has Lebesgue measure zero.
Practice 5: Almost Everywhere Equality
Define
\[ f(x)=0 \]for all \(x\), and
\[ g(x)= \begin{cases} 7, & x\in\mathbb{Z},\\ 0, & x\notin\mathbb{Z}. \end{cases} \]Are \(f\) and \(g\) equal everywhere? Are they equal almost everywhere with respect to Lebesgue measure?
Practice 6: Measurable Function
Explain why every polynomial
\[ p:\mathbb{R}\to\mathbb{R} \]is Borel measurable.
Practice 7: Indicator Function
Let
\[ E=(0,2]\cup\{5\}. \]Determine whether
\[ \mathbf{1}_E \]is Borel measurable.
Practice 8: Continuity from Below
Let
\[ E_n=[-n,n]. \]Compute
\[ \lim_{n\to\infty}m(E_n) \]and compare it with
\[ m\left( \bigcup_{n=1}^{\infty}E_n \right). \]Practice 9: Simple Function
Write the function
\[ \phi(x)= \begin{cases} 3, & x\in[0,1],\\ 1, & x\in(1,4],\\ 0, & \text{otherwise} \end{cases} \]as a linear combination of indicator functions.
Practice 10: Identify the Error
A student says:
The rational numbers are dense in \(\mathbb{R}\), so they must have positive Lebesgue measure in every interval.
Explain the error.
Practice Answers
Practice 1
Yes. The whole space \(X\) is present. The complement of \(\{1\}\) is \(\{2,3\}\), and the complement of \(\{2,3\}\) is \(\{1\}\). The collection is also closed under countable unions because there are only four possible sets.
Therefore,
\[ \boxed{\mathcal{M}\text{ is a sigma-algebra}}. \]Practice 2
The finite set has four elements, so
\[ \boxed{\mu(\{2,4,6,8\})=4}. \]The set of positive even integers is infinite, so
\[ \boxed{\mu(\{2,4,6,\ldots\})=\infty}. \]Practice 3
Since \(0\in[-1,1]\),
\[ \boxed{\delta_0([-1,1])=1}. \]Since \(0\notin(1,3)\),
\[ \boxed{\delta_0((1,3))=0}. \]Practice 4
Every singleton has Lebesgue measure zero. A finite set is a finite union of singletons, so finite additivity gives measure zero.
Practice 5
The functions are not equal everywhere because they differ at every integer.
However, \(\mathbb{Z}\) is countable, so
\[ m(\mathbb{Z})=0. \]Therefore,
\[ \boxed{f=g\text{ a.e.}}. \]Practice 6
Every polynomial is continuous on \(\mathbb{R}\). Every continuous real-valued function is Borel measurable. Therefore every polynomial is Borel measurable.
Practice 7
The interval \((0,2]\) is Borel, and the singleton \(\{5\}\) is closed and therefore Borel. Their union \(E\) is Borel measurable.
Hence
\[ \boxed{\mathbf{1}_E\text{ is Borel measurable}}. \]Practice 8
We have
\[ m(E_n)=2n. \]Therefore,
\[ \lim_{n\to\infty}m(E_n)=\infty. \]Also,
\[ \bigcup_{n=1}^{\infty}[-n,n]=\mathbb{R}, \]and
\[ m(\mathbb{R})=\infty. \]Thus continuity from below gives matching values.
Practice 9
One representation is
\[ \boxed{ \phi = 3\mathbf{1}_{[0,1]} + \mathbf{1}_{(1,4]} }. \]Practice 10
Density and measure describe different properties. The rationals are dense, meaning every nonempty open interval contains rational numbers. But \(\mathbb{Q}\) is countable, so
\[ \boxed{m(\mathbb{Q})=0}. \]A set can therefore be topologically dense while having measure zero.
What to Remember
A sigma-algebra is closed under complements and countable unions:
\[ \boxed{ (X,\mathcal{M}) \text{ is a measurable space} }. \]A measure assigns nonnegative size and is countably additive on disjoint measurable sets:
\[ \boxed{ \mu\left( \bigcup_{n=1}^{\infty}E_n \right) = \sum_{n=1}^{\infty}\mu(E_n) } \]for pairwise disjoint \(E_n\).
A measurable function preserves measurable structure through inverse images:
\[ \boxed{ f^{-1}(B)\in\mathcal{M} } \]for every Borel set \(B\).
Null sets satisfy
\[ \mu(N)=0, \]and statements that may fail only on null sets are said to hold almost everywhere.
Simple functions
\[ \phi=\sum_{k=1}^{n}a_k\mathbf{1}_{E_k} \]provide the basic building blocks from which the Lebesgue integral is constructed.
Where We Go Next
Measure theory gives us the language of measurable sets, measurable functions, null sets, and simple approximations. The next step is to turn those structures into an integral.
The next lesson, Lebesgue Integration, begins with the integral of simple functions, extends the construction to nonnegative measurable functions, and then develops integrable signed functions together with the major convergence theorems that make the Lebesgue integral central to modern analysis.