Dominated convergence theorem
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In measure theory, a branch of mathematical analysis, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and pointwise convergence for a sequence of functions. This theorem shows the superiority of the Lebesgue integral over the Riemann integral for many theoretical purposes.
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[edit] Statement of the theorem
Let f1, f2, f3, ... denote a sequence of real-valued measurable functions on a measure space (S,Σ,μ). Assume that the sequence converges pointwise and is dominated by some integrable function g. Then the pointwise limit is an integrable function and
To say that the sequence is "dominated" by g means that
for all natural numbers n and all points x in S. By integrable we mean
The convergence of the sequence and domination by g can be relaxed to hold only μ-almost everywhere.
[edit] Proof of the theorem
Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below is a direct proof, using Fatou's lemma as the essential tool.
If f denotes the pointwise limit of the sequence, then f is also measurable and dominated by g, hence integrable. Furthermore,
for all n and
By the reverse Fatou lemma,
Using linearity and monotonicity of the Lebesgue integral,
and the theorem follows.
[edit] Discussion of the assumptions
That the assumption that the sequence is dominated by some integrable g can not be dispensed with may be seen as follows: define fn(x) = n for x in the interval (0,1/n] and fn(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = supn fn. Observe that
![\int_0^1 h(x)\,dx
\ge\int_{1/m}^1 h(x)\,dx
=\sum_{n=1}^{m-1}\int_{\left(\frac1{n+1},\frac1n\right]}n\,dx
=\sum_{n=1}^{m-1}\frac1{n+1}
\to\infty\quad\text{as }m\to\infty](../../../../math/1/9/4/194ea08a721dd5b0f8591832a1d816f2.png)
by the divergence of the harmonic series. Hence, the monotonicity of the Lebesgue integral tells us that there exists no integrable function which dominates the sequence on [0,1]. A direct calculation shows that integration and pointwise limit do not commute for this sequence:
because the pointwise limit of the sequence is the zero function.
[edit] Extensions
The theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above.
[edit] See also
- Bounded convergence theorem
- Convergence in mean
- Monotone convergence theorem (does not require domination by an integrable function but assumes monotonicity of the sequence instead)
- Scheffé's lemma
- Uniform integrability
- Vitali convergence theorem (a generalization of Lebesgue's dominated convergence theorem)
[edit] References
- R.G. Bartle, "The Elements of Integration and Lebesgue Measure", Wiley Interscience, 1995.
- H.L. Royden, "Real Analysis", Prentice Hall, 1988.
- D. Williams, "Probability with Martingales", Cambridge University Press, 1991, ISBN 0-521-40605-6
