Integration and Measure Theory

Intuition: A partition chops $[a, b]$ into finitely many subintervals. A refinement adds more dividing points, so every subinterval of $P'$ sits inside a subinterval of $P$. Finer partitions give finer approximations of area.

Intuition: The upper sum is the area of the smallest "staircase" that fits above the graph of $f$, while the lower sum is the area of the largest staircase fitting below. A finer partition can only make $U$ smaller and $L$ larger, squeezing them toward the "true" area.

Intuition: The upper integral $\inf_P U(f, P)$ is the best upper estimate for the area under $f$; the lower integral $\sup_P L(f, P)$ is the best lower estimate. When they agree, there is a unique sensible notion of area, and we call the common value the Riemann integral. From the lemma above, we always have lower integral $\leq$ upper integral.

Intuition: Riemann integrability is equivalent to the gap between upper and lower sums being controllable. This "squeeze" criterion is the workhorse for proving integrability in practice.

Intuition: These properties are the bedrock of calculus. Linearity and monotonicity follow directly from the definition of Darboux sums; additivity is proved by concatenating partitions on $[a, c]$ and $[c, b]$; and absolute integrability uses $||f(x)| - |f(y)|| \leq |f(x) - f(y)|$ to control oscillations of $|f|$ by oscillations of $f$.

Intuition: To integrate on a general bounded set, we set $f$ to be zero outside and integrate over any enclosing interval. The choice of enclosing interval does not matter because extending the partition outside $A$ only adds zero contributions.

Intuition: Jordan content is the Riemann theorist's notion of "length" (or "area" in higher dimensions). A set is Jordan measurable exactly when its characteristic function is Riemann integrable, i.e. when upper and lower sums of $\mathbf{1}_A$ can be made arbitrarily close. Intervals $[a, b]$ have Jordan content $b - a$; finite sets have Jordan content $0$.

Intuition: Jordan content zero sets can be covered by a finite collection of intervals of arbitrarily small total length. This finiteness is the crucial limitation: it rules out sets like $\mathbb{Q} \cap [0, 1]$, which would require countably (not finitely) many intervals.

Intuition: A set has zero Lebesgue measure if it can be covered by a countable collection of intervals of arbitrarily small total length. The difference from Jordan content zero is that we now allow infinitely many intervals, which is far more flexible: countable sets always have Lebesgue measure zero, even when they have no Jordan content.

Jordan content zero implies zero Lebesgue measure. If $A$ has Jordan content zero, then the finite interval covers in the Jordan definition form (in particular) countable covers, so $A$ has zero Lebesgue measure. The converse is false in general, as $\mathbb{Q} \cap [0, 1]$ shows. However, for compact sets the two notions coincide: a compact set has Jordan content zero iff it has zero Lebesgue measure.

Intuition: This is the first essential advantage of Lebesgue measure over Jordan content: Jordan content is not closed under countable unions. Countable additivity of Lebesgue measure is what will eventually make the Lebesgue integral so much more flexible.

Intuition: This is the cleanest possible description of which bounded functions are Riemann integrable. Continuity is not necessary; being almost everywhere continuous — with the "bad" set being negligible in the measure sense — is exactly what's needed. For example, the Dirichlet function is discontinuous on all of $[0, 1]$, which is a set of Lebesgue measure $1 \neq 0$, so it is not Riemann integrable. A bounded function continuous except on a countable set is Riemann integrable.

Intuition: A $\sigma$-algebra is a collection of sets that is closed under complementation and countable unions. Together with $\emptyset, X$, this closure makes it stable under every countable set-theoretic operation: finite and countable intersections, differences, symmetric differences, limsup/liminf of sequences of sets, and so on. It is exactly the minimal structure needed to consistently assign "sizes" via countable operations.

Intuition: $\sigma(\mathcal{G})$ is the smallest $\sigma$-algebra containing $\mathcal{G}$. It exists because $2^X$ itself is always a $\sigma$-algebra containing $\mathcal{G}$, so the collection being intersected is nonempty, and the intersection is a $\sigma$-algebra by the previous theorem.

Intuition: Borel sets are the sets you can build from open sets by applying countable unions, countable intersections, and complements (possibly repeatedly). Every open set is Borel, every closed set is Borel, every countable union of closed sets (an $F_\sigma$) is Borel, every countable intersection of open sets (a $G_\delta$) is Borel, and so on. For practical purposes, every set you can "describe reasonably" on $\mathbb{R}$ is Borel.

Intuition: Any of these natural collections of intervals generates the Borel $\sigma$-algebra. The essential observation is that open intervals, closed intervals, and half-open intervals can all be built from one another using countable operations (e.g. $[a, b] = \bigcap_n (a - 1/n, b + 1/n)$ and $(a, b) = \bigcup_n [a + 1/n, b - 1/n]$).

Intuition: A measure is a consistent way of assigning a nonnegative size to each measurable set, respecting the fact that disjoint pieces should have sizes that add. Countable additivity is the key axiom: it handles limit operations correctly. From countable additivity one derives $\mu(\emptyset) = 0$ (using any $A$ with $\mu(A) < \infty$ and the disjoint union $A = A \cup \emptyset \cup \emptyset \cup \cdots$).

Intuition: An outer measure is a set function defined on every subset that is monotone and subadditive over countable unions. It is not a measure in general — there is no additivity, only subadditivity. The point is that outer measures are easy to construct, and we can restrict to measurable sets to recover genuine additivity.

Intuition: The Lebesgue outer measure of $A$ is the infimum of total lengths of countable covers of $A$ by half-open intervals. This is exactly the definition of "zero Lebesgue measure" from earlier — that case corresponds to $\mu^*(A) = 0$.

Intuition: The Caratheodory criterion says that $A$ cleanly "splits" every test set $E$ into two pieces whose outer measures add. By countable subadditivity, the inequality $\mu^*(E) \leq \mu^*(E \cap A) + \mu^*(E \cap A^c)$ is automatic; the content is the reverse inequality. The criterion singles out sets that are nice enough for additivity to hold.

Intuition: Caratheodory's theorem is the engine that produces measures out of outer measures. Starting from any outer measure, we obtain a $\sigma$-algebra (automatically closed under all countable operations) and a genuine measure on it. The trade-off is that $\mathcal{M}$ may be smaller than $2^X$: not every set is Caratheodory measurable.

Intuition: Lebesgue measure is the unique "natural" extension of length from intervals to a large class of subsets of $\mathbb{R}$, obtained by combining the covering-class outer measure construction with the Caratheodory restriction. It is countably additive, assigns length $b - a$ to the interval $[a, b)$, and contains every Borel set.

Intuition: These are the basic features that justify calling $\mu$ "length": it agrees with usual length on intervals, it contains all Borel sets (so in particular every set we can "describe"), it is countably additive, and it is invariant under translation — the geometry of the real line is respected. Countable additivity makes it vastly more flexible than Jordan content, and countable unions of null sets remain null, which is the technical underpinning of the Lebesgue integral.

Why not just use Borel sets? The Lebesgue $\sigma$-algebra is strictly larger than the Borel $\sigma$-algebra; it contains, for instance, every subset of a Lebesgue null Borel set (such subsets need not be Borel). This completeness property is useful because it implies that changing an integrable function on a null set does not affect measurability or integrability.

Intuition: Covering $[a, b)$ by itself gives $\mu^*([a, b)) \leq b - a$. The nontrivial direction is that every countable cover of $[a, b)$ by half-open intervals has total length $\geq b - a$; this is proved by a compactness argument (reduce to finite covers using Heine–Borel after a small $\varepsilon$-enlargement to open intervals, then add up lengths along the interval). So the outer measure truly extends length.

Intuition: Since the Lebesgue $\sigma$-algebra $\mathcal{M}$ is a $\sigma$-algebra that contains every interval (via the Caratheodory criterion applied to half-open intervals), it must contain the $\sigma$-algebra generated by intervals — which is exactly $\mathcal{B}(\mathbb{R})$. Hence open sets, closed sets, $F_\sigma$'s, $G_\delta$'s, and so on, are all Lebesgue measurable.

Intuition: Measurability is exactly the same as "approximable above by an open set and below by a closed set, to arbitrary precision." This is the technical underpinning of many constructions in integration: we can always replace a measurable set by an open or closed approximation and control the error by the measure of the symmetric difference.

Intuition: The explicit construction (due to Vitali) uses the axiom of choice to pick one representative from each coset of $\mathbb{Q}$ inside $[0, 1]$. The resulting "Vitali set" cannot be assigned a Lebesgue measure consistent with translation invariance and countable additivity. The proof is outside the main development, but the take-away is important: extending "length" to every subset of $\mathbb{R}$ in a way compatible with countable additivity and translation invariance is impossible. Lebesgue's $\sigma$-algebra is as large as we can hope for without abandoning these properties.

Intuition: Simple functions are the measure-theoretic analogue of step functions, with arbitrary measurable level sets replacing intervals. Their integral is defined by "area = value times measure of level set," summed over level sets. Every measurable function is a monotone limit of simple functions, so Lebesgue integration of general measurable functions is built by extending this definition through limits.

Intuition: This mirrors the Riemann sum $\sum_k f(\xi_k) (x_k - x_{k-1})$, but instead of partitioning the domain into intervals, we partition the range into level values and weight each by the measure of the preimage. This "horizontal slicing" is what makes Lebesgue integration so flexible: it makes sense whenever the preimages are measurable, regardless of how wildly they are scattered in the domain.

Intuition: This example is the cleanest illustration of the superiority of the Lebesgue theory. Changing an integrable function on a null set does not affect its Lebesgue integral; in particular, functions that agree almost everywhere have the same integral. Riemann integration lacks this property because the Riemann integral is too closely tied to the continuity of the function at individual points rather than to its global behaviour on the level sets.

Beyond simple functions. The Lebesgue integral can be extended to a much wider class of functions — the measurable functions — by approximating them as monotone (or dominated) limits of simple functions and taking the limits of the corresponding integrals. This is developed in the standard reference (Terrell, Chapter 17), and yields the full-fledged Lebesgue integral, along with its powerful convergence theorems (monotone convergence, dominated convergence, Fatou's lemma). For the purposes of this course, the key ideas are: (i) what measurable sets are, (ii) that Lebesgue measure genuinely extends length, and (iii) that simple functions can already integrate examples out of reach of the Riemann theory.