If lnx is integrable, then is xlnx also integrable?I have a very simple problem. Assume we have a finite measure μ on [ 1 , ∞ ), and ∫ 1 ∞ t d μ ( t ) &lt; ∞ . My question is if this implies ∫ 1 ∞ t ln ⁡ t d μ ( t ) &lt; ∞ . It would be nice if this is true but maybe someone sees a simple counterexample?

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If lnx is integrable, then is xlnx also integrable?I have a very simple problem. Assume we have a finite measure   μ on   [  1  ,  ∞  ), and                              ∫          1          ∞                t                 d        μ        (        t        )        &amp;lt;        ∞        .            My question is if this implies                              ∫          1          ∞                t        ln        ⁡        t                 d        μ        (        t        )        &amp;lt;        ∞        .            It would be nice if this is true but maybe someone sees a simple counterexample?

wespee0 · Accepted Answer

Example:   d  μ  (  t  )  =                    d        t                              t          2                          log                      3                          /                        2                          ⁡        (        t        +        1        )

Alisa Taylor · Accepted Answer

Let       d    μ  =                    d            t                      t        α                    ln        β            ⁡      t      . You should be able to show the two integrals converge if   α  &amp;gt;  2 and, in the first case,   β  &amp;lt;  1, and in the second,   β  &amp;lt;  2. Surely you can find something in   (  1  ,  2  ].

If ln x is integrable, then is x ln x also integrable? I have a very simple problem. Assume we have a finite measure mu on [1,oo), and int_1^oo t d mu(t) < oo.

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