Real Analysis: Measure and Integration

Modern real analysis treats integration not as a clever limit of areas under curves, but as a systematic way to measure the size of sets and accumulate values of functions. This shift is what makes Lebesgue’s theory so powerful. It replaces the Riemann integral with a framework that handles discontinuities cleanly, supports robust limit operations, and scales naturally to higher dimensions and probability.

At the center of this subject are four pillars: Lebesgue measure, measurable functions, convergence theorems, and Fubini’s theorem. Together they form the working language of analysis, partial differential equations, Fourier analysis, and modern probability.

Why move beyond Riemann integration?

Riemann integration partitions the domain into intervals and approximates the integral by sampling function values. This works well for continuous functions and many piecewise-continuous ones, but it struggles when discontinuities are widespread. A classic limitation is that a bounded function on an interval is Riemann integrable precisely when its set of discontinuities has measure zero, a condition that excludes many naturally occurring examples.

Lebesgue’s idea is to reverse the viewpoint. Instead of chopping up the domain, you classify points according to the function’s values and measure the size of the corresponding level sets. This approach makes it possible to integrate highly irregular functions, and it makes limit processes far more reliable.

Lebesgue measure: assigning length in a stable way

Lebesgue measure extends the intuitive notion of length (and, in higher dimensions, area and volume) to a very large collection of sets, while preserving the properties that make “length” useful.

Core properties

On $\mathbb{R}$, Lebesgue measure $m(\cdot )$ is characterized (informally) by:

• Translation invariance: $m(A+x)=m(A)$ for any set $A$ and shift $x$.
• Countable additivity: if $A_1,A_2,\dots$ are disjoint measurable sets, then

$m\left({\bigcup }_{n=1}^{\infty }{A}_n\right)={\sum }_{n=1}^{\infty }m(A_n)$.

• Agreement with intervals: $m((a,b))=b-a$ and similarly for other interval types.

Countable additivity is the decisive upgrade from finite additivity. It allows one to control infinite decompositions and is essential for convergence theorems later on.

Measurable sets and why not everything is measurable

Lebesgue measure is defined on the Lebesgue measurable sets, a collection rich enough to include all Borel sets (sets built from open sets via countable unions and intersections) and many more. Not every subset of $\mathbb{R}$ can be measured consistently while keeping translation invariance and countable additivity; this is why “measurable” is a nontrivial requirement rather than a technicality.

A set of measure zero, such as a countable set, is negligible from the viewpoint of integration. This idea underlies the phrase “almost everywhere” (a.e.): a property holds almost everywhere if it fails only on a set of measure zero.

Measurable functions: the right class for integration

To integrate functions in the Lebesgue sense, we need to know that sets defined by comparing the function to real numbers are measurable.

Definition and intuition

A function $f:\mathbb{R}\to \mathbb{R}$ is measurable if for every real $a$, the set $\{x:f(x)>a\}$ is measurable. Equivalent formulations use $\ge a$, $<a$, or preimages of open sets.

Measurable functions include:

• all continuous functions,
• all monotone functions,
• limits of measurable functions under common modes of convergence,
• characteristic functions $1_A$ of measurable sets.

The ability to build complicated functions from simple ones while keeping measurability is one reason the theory is so flexible.

Lebesgue integration: from simple functions to general functions

Lebesgue integration is constructed in stages, which keeps the definition grounded and makes the major theorems believable.

Step 1: simple functions

A simple function has the form \[ \phi = \sum{k=1}^n ck \mathbf{1}{Ak}, \] where the $A_k$ are measurable and the $c_k$ are real constants. Its integral is defined by \[ \int \phi \, dm = \sum{k=1}^n ck\, m(A_k). \] This matches the intuition “value times size,” summed over regions where the function is constant.

Step 2: nonnegative measurable functions

For a nonnegative measurable $f$, define \[ \int f\, dm = \sup\left\{\int \phi\, dm : 0 \le \phi \le f,\ \phi \text{ simple}\right\}. \] So the integral is the best possible approximation from below by simple functions. This is the counterpart to lower sums, but it is designed to behave well under limits.

Step 3: integrable signed functions

For general measurable $f$, write $f=f^{+}-f^{-}$ where $f^{+}=\max (f,0)$ and $f^{-}=\max (-f,0)$. If both $\int f^{+}dm$ and $\int f^{-}dm$ are finite, then $f$ is (Lebesgue) integrable and \[ \int f\, dm = \int f^+ dm - \int f^- dm. \]

Convergence theorems: the engine of the subject

The most practical advantage of Lebesgue integration is that it provides clean conditions under which you can interchange limits and integrals. This is essential in analysis, where functions are often defined as limits or approximated by sequences.

Monotone Convergence Theorem (MCT)

If $0\le f_n\uparrow f$ pointwise (that is, $f_n(x)$ increases to $f(x)$ for every $x$), then \[ \int f_n \, dm \to \int f \, dm. \] This theorem is almost built into the definition of the integral as a supremum of simple approximations. It is widely used when constructing functions by increasing approximations, such as truncations or stepwise limits.

Fatou’s lemma

If $f_n\ge 0$ are measurable, then \[ \int \liminf{n\to\infty} fn \, dm \le \liminf{n\to\infty} \int fn \, dm. \] Fatou’s lemma is a one-sided inequality that often provides bounds when full limit interchange is not yet justified.

Dominated Convergence Theorem (DCT)

If $f_n\to f$ almost everywhere and there exists an integrable function $g$ such that $|f_n|\le g$ a.e. for all $n$, then $f$ is integrable and \[ \int fn \, dm \to \int f \, dm. \] DCT is the workhorse theorem in applications. In practice, the challenge is to find a suitable dominating function MATHINLINE35. Common choices come from uniform bounds, integrable tails, or comparison with known MATHINLINE36_ functions.

A practical pattern

A typical application looks like this: define $f_n(x)=f(x)1_{[-n,n]}(x)$ to truncate an unbounded domain. If $f$ is integrable, then $|f_n|\le |f|$ and $f_n\to f$ a.e., so DCT yields $\int f_n\to \int f$. This kind of argument is routine in Fourier analysis and probability.

Fubini and Tonelli: integrating over products

When working on ${\mathbb{R}}^2$ or higher dimensions, you often want to compute integrals by iterated integration. Fubini’s theorem and its nonnegative counterpart, Tonelli’s theorem, formalize when this is valid.

Tonelli’s theorem (nonnegative case)

If $f\ge 0$ is measurable on $X\times Y$, then \[ \int{X\times Y} f \, d(mX \times mY) = \intX\left(\intY f(x,y)\, dmY(y)\right) dmX(x), \] and the iterated integrals may take the value MATHINLINE45_. Tonelli is powerful because it requires no integrability assumption beyond nonnegativity.

Fubini’s theorem (integrable case)

If $f$ is integrable on $X\times Y$ (that is, $\int |f|<\infty$), then the iterated integrals exist as finite numbers for almost every slice and \[ \int{X\times Y} f = \intX\left(\intY f(x,y)\, dy\right) dx = \intY\left(\int_X f(x,y)\, dx\right) dy. \] Fubini is what justifies switching the order of integration in legitimate calculations, such as evaluating integrals by choosing the easier order or proving identities involving convolutions.

Why the hypotheses matter

Order switching can fail for functions that are not absolutely integrable. Lebesgue theory makes