Lebesgue's dominated convergence theorem

Dominated convergence theorem

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.

Statement of the theorem

Let f1, f2, f3, ... denote a sequence of real-valued measurable functions on a measure space (S,Sigma,mu). Assume that the sequence converges pointwise and is dominated by some integrable function g. Then the pointwise limit is an integrable function and
int_Slim_{ntoinfty} f_n,dmu=lim_{ntoinfty}int_S f_n,dmu. To say that the sequence is "dominated" by g means that
|f_n(x)| le g(x) for all natural numbers n and all points x in S. By integrable we mean
int_S|g|,dmu

The convergence of the sequence and domination by g can be relaxed to hold only mu-almost everywhere.

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,

|f-f_n|le 2g for all n and
limsup_{ntoinfty}|f-f_n|=0. By the reverse Fatou lemma,
limsup_{ntoinfty}int_S|f-f_n|,dmu leint_Slimsup_{ntoinfty}|f-f_n|,dmu=0. Using linearity and monotonicity of the Lebesgue integral,
biggl|int_Sf,dmu-int_Sf_n,dmubiggr| =biggl|int_S(f-f_n),dmubiggr| leint_S|f-f_n|,dmu, and the theorem follows.

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 geint_{1/m}^1 h(x),dx

sum_{n

1}^{m-1}int_{left(frac1{n+1},frac1nright]}n,dx

sum_{n

1}^{m-1}frac1{n+1} toinftyquadtext{as }mtoinfty

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:

int_0^1lim_{ntoinfty} f_n(x),dx

0neq 1

lim_{ntoinfty}int_0^1 f_n(x),dx,

because the pointwise limit of the sequence is the zero function.

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.

See also

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

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