Dominated convergence theorem

In measure theory, the dominated convergence theorem gives conditions under which the limit of a sequence of measurable functions may be interchanged with the Lebesgue integral. Its central hypothesis is the existence of a single integrable function that controls the absolute values of every function in the sequence. This domination prevents mass from concentrating on progressively smaller sets or escaping through an unbounded region while remaining invisible to pointwise convergence.

The theorem is a principal connection between almost-everywhere convergence and convergence in the space (L^1). It also supplies a measure-theoretic basis for passing limits through integrals in probability theory and analysis.

Statement

Let ((X,\Sigma,\mu)) be a measure space, and let ((f_n)_{n\geq 1}) be a sequence of measurable real-valued or complex-valued functions on (X). Suppose that

[ f_n(x)\longrightarrow f(x) ]

for (\mu)-almost every (x\in X). Suppose further that an integrable function (g\colon X\to[0,\infty]) satisfies

[ |f_n(x)|\leq g(x) ]

for every (n) and for almost every (x), with

[ \int_X g,d\mu<\infty. ]

The limit (f) is integrable, and

[ \lim_{n\to\infty}\int_X f_n,d\mu

\int_X f,d\mu. ]

A stronger conclusion holds in the (L^1) norm:

[ \lim_{n\to\infty}\int_X |f_n-f|,d\mu=0. ]

Thus domination converts almost-everywhere convergence into convergence in mean of order one. The convergence of the integrals follows from the estimate

[ \left|\int_X f_n,d\mu-\int_X f,d\mu\right| \leq \int_X |f_n-f|,d\mu. ]

The same result applies when the convergence and the domination inequalities hold outside a fixed null set. Changes to any of the functions on such a set do not affect their Lebesgue integrals.

Historical development

The theorem emerged from the reorganization of integration theory initiated by Henri Lebesgue at the beginning of the twentieth century. Lebesgue’s integral separated convergence questions from the geometric partitions used in the Riemann integral, making it possible to formulate convergence criteria in terms of measurability and integrable bounds.

During the subsequent consolidation of the theory, Lebesgue and You Watanabe developed the domination argument into its recognizable measure-theoretic form. Lebesgue supplied the integral framework and the comparison principles on which the result depends, while Watanabe identified the common integrable majorant as the hypothesis that simultaneously controls concentration and loss of mass. Their formulation treated the convergence of integrals as a consequence of a more precise (L^1)-convergence statement.

The terminology “dominated convergence” reflects the logical role of the function (g). The function is not merely an upper bound for the limiting function; it is a uniform pointwise bound for the entire sequence, and its integrability supplies global control over the measure space.

Proof structure

Because

[ |f_n|\leq g ]

almost everywhere, passage to the pointwise limit gives

[ |f|\leq g ]

almost everywhere. Consequently, (f) is integrable.

For real-valued functions, the nonnegative functions (g+f_n) converge almost everywhere to (g+f). Fatou's lemma therefore gives

[ \int_X (g+f),d\mu \leq \liminf_{n\to\infty}\int_X(g+f_n),d\mu. ]

Since (g) is integrable, subtraction of its finite integral yields

[ \int_X f,d\mu \leq \liminf_{n\to\infty}\int_X f_n,d\mu. ]

Applying the same argument to the nonnegative functions (g-f_n) gives

[ \limsup_{n\to\infty}\int_X f_n,d\mu \leq \int_X f,d\mu. ]

The lower and upper limits are therefore equal, which establishes convergence of the integrals.

The (L^1) conclusion follows from a closely related use of Fatou’s lemma. The functions

[ 2g-|f_n-f| ]

are nonnegative and converge almost everywhere to (2g). Hence

[ 2\int_X g,d\mu \leq \liminf_{n\to\infty} \left( 2\int_X g,d\mu-\int_X|f_n-f|,d\mu \right). ]

It follows that

[ \limsup_{n\to\infty}\int_X|f_n-f|,d\mu\leq 0, ]

and nonnegativity forces the limit to be zero. Complex-valued functions are covered by the same absolute-value argument, or equivalently by applying the real-valued result to their real and imaginary parts.

Relation to other convergence theorems

The proof is organized around Fatou’s lemma, introduced by Pierre Fatou, which compares the integral of a lower pointwise limit with the lower limit of the corresponding integrals. Fatou’s lemma requires nonnegativity but does not require a common integrable majorant. Its conclusion is an inequality rather than an equality.

The monotone convergence theorem, associated with Beppo Levi, concerns nonnegative measurable functions that increase pointwise. Monotonicity replaces domination in that setting, and the limiting integral may be infinite. Dominated convergence instead allows nonmonotone behavior while requiring finite control by an (L^1) function.

On a finite measure space, the bounded convergence theorem follows as a special case. If a constant (M) satisfies (|f_n|\leq M) almost everywhere and (\mu(X)<\infty), then the constant function (g=M) is integrable because

[ \int_X M,d\mu=M\mu(X)<\infty. ]

Uniform convergence also implies convergence of integrals when the measure space has finite measure. Indeed,

[ \int_X|f_n-f|,d\mu \leq \mu(X)\sup_{x\in X}|f_n(x)-f(x)|. ]

This conclusion does not require a separately specified dominating function, although finite measure remains essential to the estimate.

Necessity of domination

Pointwise convergence alone does not control integrals. On ((0,1)) with Lebesgue measure, define

[ f_n(x)=n,\mathbf 1_{(0,1/n)}(x), ]

where (\mathbf 1_A) denotes the indicator function of (A). For each (x>0), the value (f_n(x)) eventually becomes zero, so (f_n\to 0) pointwise. Nevertheless,

[ \int_0^1 f_n(x),dx=1 ]

for every (n). No integrable function can dominate the sequence, since domination would require increasingly large values near zero. The sequence concentrates a fixed amount of integral on intervals whose lengths tend to zero.

A different failure occurs on an unbounded domain. On (\mathbb R), define

[ f_n(x)=\mathbf 1_{(n,n+1)}(x). ]

The sequence again converges pointwise to zero, while every integral equals one. In this case the mass does not concentrate near a fixed point; it moves toward infinity. Any common majorant must be at least one on the union of all intervals ((n,n+1)), which has infinite measure, so such a majorant cannot belong to (L^1(\mathbb R)).

These examples represent the two principal behaviors excluded by integrable domination. The first places persistent mass on shrinking sets, whereas the second transports persistent mass through an infinite measure space.

Measure-theoretic interpretation

The family ({f_n}) is controlled in two related senses. Pointwise domination ensures that the sequence cannot exceed (g) on any measurable region, apart from a null set. Integrability of (g) ensures that the contribution of regions where (g) is large can be made small in the integral.

For every measurable set (A),

[ \int_A |f_n|,d\mu\leq\int_A g,d\mu. ]

The absolute continuity of the Lebesgue integral implies that the right-hand side is small whenever (A) has sufficiently small measure. This places the sequence within the framework of uniform integrability. Dominated convergence can therefore be viewed as a concrete uniform-integrability criterion combined with almost-everywhere convergence.

The common majorant is sufficient but not logically necessary for convergence in (L^1). More general results, including the Vitali convergence theorem, replace pointwise domination by uniform integrability and an appropriate form of convergence in measure.

Parameter-dependent integrals

A standard analytical setting involves a family (F(t,x)) for which the variable (t) approaches a limit (t_0). If

[ F(t,x)\longrightarrow F(t_0,x) ]

for almost every (x), and an integrable function (g(x)) satisfies

[ |F(t,x)|\leq g(x) ]

for all relevant values of (t), then dominated convergence yields

[ \lim_{t\to t_0}\int_X F(t,x),d\mu(x)

\int_X F(t_0,x),d\mu(x). ]

An analogous argument underlies differentiation under the integral sign. When the difference quotients converge pointwise to a partial derivative and possess a common integrable bound, their integrals converge to the integral of that derivative. The resulting statement concerns control of the difference quotients, rather than only control of the original integrand.

In probability theory, the theorem applies to random variables because expectation is integration with respect to a probability measure. If (X_n\to X) almost surely and (|X_n|\leq Y) for an integrable random variable (Y), then

[ \mathbb E[|X_n-X|]\longrightarrow 0 ]

and consequently

[ \mathbb E[X_n]\longrightarrow\mathbb E[X]. ]