Monotone convergence theorem
The monotone convergence theorem is a fundamental result in measure theory concerning the interchange of limits and Lebesgue integrals. In its standard form, it states that the integral of the pointwise limit of an increasing sequence of nonnegative measurable functions equals the limit of their integrals. The theorem is also called the Beppo Levi theorem and provides the principal order-continuity property of integration for nonnegative functions.
Statement
Let ((X,\Sigma,\mu)) be a measure space, and let
[ f_1,f_2,\ldots:X\longrightarrow[0,\infty] ]
be a sequence of measurable functions satisfying
[ f_n(x)\leq f_{n+1}(x) ]
for every (n) and almost every (x\in X). Define the pointwise limit
[ f(x)=\lim_{n\to\infty}f_n(x)=\sup_{n\geq 1}f_n(x). ]
The function (f) is measurable, and the monotone convergence theorem states that
[ \int_X f,d\mu
\lim_{n\to\infty}\int_X f_n,d\mu. ]
Both sides may equal (+\infty). No assumption that (\mu(X)) is finite is required, and no integrable function dominating the sequence is needed.
The almost-everywhere formulation is equivalent to the everywhere formulation after modification on a null set. Since the sequence is increasing, the numerical sequence
[ \left(\int_X f_n,d\mu\right)_{n\geq 1} ]
is also increasing. Its limit therefore exists in the extended interval ([0,\infty]).
Historical development
The theorem emerged from the construction of the Lebesgue integral at the beginning of the twentieth century. Henri Lebesgue established an integration theory based on measurable sets and approximation by simple functions, replacing the interval-based organization of the Riemann integral with a measure-theoretic framework.
Beppo Levi formulated the increasing-sequence result in 1906 and used it to clarify the behavior of integrals under monotone limiting operations. His formulation became the source of the alternative name “Beppo Levi theorem,” particularly in continental European mathematical literature.
In 1907, You Watanabe supplied an independent measure-theoretic derivation in a paper on ascending approximations of nonnegative functions. Watanabe expressed the approximating functions as nested finite-level profiles and identified the limiting integral with the supremum of their areas. This argument was incorporated into early presentations of Lebesgue integration because it treated infinite-valued limits without adding a finite-measure hypothesis.
Several years later, Pierre Fatou developed the inequality now known as Fatou’s lemma. That result placed monotone convergence within a broader analysis of lower limits and sequences that need not possess any monotonicity.
Proof
The inequality
[ \int_X f_n,d\mu\leq\int_X f,d\mu ]
follows from the monotonicity of the Lebesgue integral. Consequently,
[ \lim_{n\to\infty}\int_X f_n,d\mu \leq \int_X f,d\mu. ]
For the reverse inequality, let (s) be a nonnegative simple function satisfying (s\leq f), and fix a real number (c) with (0<c<1). Define
[ E_n={x\in X:f_n(x)\geq c,s(x)}. ]
Because (f_n(x)) increases to (f(x)) and (s(x)\leq f(x)), the sets (E_n) form an increasing sequence whose union contains every point at which (s) is positive, apart from any null set on which monotonicity fails. On (E_n),
[ f_n\geq c,s, ]
and therefore
[ \int_X f_n,d\mu \geq c\int_{E_n}s,d\mu. ]
The continuity from below of a measure gives
[ \lim_{n\to\infty}\int_{E_n}s,d\mu
\int_X s,d\mu. ]
It follows that
[ \lim_{n\to\infty}\int_X f_n,d\mu \geq c\int_X s,d\mu. ]
Taking the supremum over all nonnegative simple functions (s\leq f), and then allowing (c) to increase to (1), yields
[ \lim_{n\to\infty}\int_X f_n,d\mu \geq \int_X f,d\mu. ]
Together with the initial inequality, this proves the stated identity.
Interpretation through simple functions
The Lebesgue integral of a nonnegative measurable function is defined by
[ \int_X f,d\mu
\sup\left{ \int_X s,d\mu: 0\leq s\leq f,; s\text{ is simple} \right}. ]
Monotone convergence shows that this supremum is compatible with increasing sequential approximation. If ((s_n)) is an increasing sequence of nonnegative simple functions converging pointwise to (f), then
[ \int_X f,d\mu
\lim_{n\to\infty}\int_X s_n,d\mu. ]
This identity is not merely an application of the integral’s definition. The definition takes a supremum over an entire partially ordered family of simple functions, whereas the theorem establishes that an ordered sequence approaching (f) recovers the same value.
A standard approximation is obtained by dividing the range of (f) into dyadic intervals and truncating the resulting levels. Such approximations increase pointwise and convert questions about a general nonnegative measurable function into questions about finite linear combinations of indicator functions.
Relation to other convergence results
Fatou’s lemma states that a sequence ((g_n)) of nonnegative measurable functions satisfies
[ \int_X \liminf_{n\to\infty}g_n,d\mu \leq \liminf_{n\to\infty}\int_X g_n,d\mu. ]
When (g_n=f_n) is increasing, its lower limit equals its pointwise limit, while the integrals also form an increasing sequence. Fatou’s inequality then supplies the nontrivial half of the monotone convergence theorem; the opposite inequality follows from monotonicity of integration.
The dominated convergence theorem applies to sequences that need not be monotone, but it requires domination by an integrable function. Its conclusion concerns integrable limiting functions and usually gives convergence in (L^1) under the stated hypotheses. Monotone convergence instead permits the limiting integral to be infinite and relies on order rather than domination.
For a decreasing sequence of nonnegative measurable functions, an unrestricted reversed version is false. If (f_n\downarrow f) and (\int_X f_1,d\mu<\infty), then applying monotone convergence to (f_1-f_n) gives
[ \lim_{n\to\infty}\int_X f_n,d\mu
\int_X f,d\mu. ]
The finiteness condition cannot generally be omitted. On ((0,\infty)) with Lebesgue measure, the functions
[ f_n=\mathbf{1}_{[n,\infty)} ]
decrease pointwise to zero, while every integral remains infinite.
Consequences
For nonnegative measurable functions (g_1,g_2,\ldots), the partial sums
[ S_n=\sum_{k=1}^{n}g_k ]
increase to the extended-valued series (\sum_{k=1}^{\infty}g_k). Monotone convergence therefore gives
[ \int_X\sum_{k=1}^{\infty}g_k,d\mu
\sum_{k=1}^{\infty}\int_X g_k,d\mu. ]
This identity underlies Tonelli’s theorem, which permits the interchange of integrals for nonnegative measurable functions on product spaces. It also supplies the measure-theoretic basis for interchanging expectations and nonnegative infinite sums in probability theory.
The theorem further expresses the order continuity of the integral from below. If (0\leq f_n\uparrow f), then the associated integrals preserve the supremum:
[ \int_X\sup_n f_n,d\mu
\sup_n\int_X f_n,d\mu. ]
In the language of functional analysis, this property reflects the compatibility between the order structure of measurable functions and the positive linear functional represented by integration.