Dominated Convergence Theorem - Discussion of The Assumptions

Discussion of The Assumptions

The assumption that the sequence is dominated by some integrable g can not be dispensed with. This may be seen as follows: define ƒn(x) = n for x in the interval (0, 1/n] and ƒn(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = supn ƒn. 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} \frac{1}{n+1} \to \infty \quad \text{as }m\to\infty

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 . A direct calculation shows that integration and pointwise limit do not commute for this sequence:

\int_0^1 \lim_{n\to\infty} f_n(x)\,dx = 0 \neq 1 = \lim_{n\to\infty}\int_0^1 f_n(x)\,dx,

because the pointwise limit of the sequence is the zero function. Note that the sequence {ƒn} is not even uniformly integrable, hence also the Vitali convergence theorem is not applicable.

Read more about this topic:  Dominated Convergence Theorem

Famous quotes containing the words discussion of, discussion and/or assumptions:

    “Americans, unhappily, have the most remarkable ability to alchemize all bitter truths into an innocuous but piquant confection and to transform their moral contradictions, or public discussion of such contradictions, into a proud decoration, such as are given for heroism on the battle field.”
    —James Baldwin (1924–1987)

    “What chiefly distinguishes the daily press of the United States from the press of all other countries is not its lack of truthfulness or even its lack of dignity and honor, for these deficiencies are common to the newspapers everywhere, but its incurable fear of ideas, its constant effort to evade the discussion of fundamentals by translating all issues into a few elemental fears, its incessant reduction of all reflection to mere emotion. It is, in the true sense, never well-informed.”
    —H.L. (Henry Lewis)

    “Assumptions of male superiority are as widespread and deep rooted and every bit as crippling to the woman as the assumptions of white supremacy are to the Negro.... this is no more a man’s world than it is a white world.”
    —Student Non-Violent Coordinating Committee, African American civil rights organization. SNCC Position Paper (Women in the Movement)