Does lim n → ∞ ∫ a b f ( x ) p n ( x ) d x = ∫ a b f ( x ) p ( x ) d x hold in general?

Question

Does                               lim                      n            →            ∞                                    ∫          a          b                f        (        x                  )                                    p              n                        (            x            )                          d        x        =                  ∫          a          b                f        (        x                  )                      p            (            x            )                          d        x             hold in general?

getrdone07tl · Accepted Answer

The answer is negative: On [0,1] define   p  ≡  2  ,      p    n    (  x  )  =  2  +      χ          (      1              /            n      ,      1              /            (      n      −      1      )      )        ,  n  =  2  ,  3  ,  … Then       p    n    →  p in       L    1    .Now define      f    n    (  x  )  =      1          2      n                  χ              (        1                  /                n        ,        1                  /                (        n        −        1        )        )                            x        −        1                  /                n                    |            ln      ⁡      (      x      −      1              /            n      )              |            and put   f  =  ∑      f    n    . Then       ∫    0    1        f    2    &amp;lt;  ∞  . But for each n      ∫    0    1        f                  p        n              ≥      ∫          1              /            n              1              /            (      n      −      1      )            f    n    3    =      1          2              3        n                  ∫          1              /            n              1              /            (      n      −      1      )            1          (      x      −      1              /            n              )                  3                      /                    2                            |            ln      ⁡      (      x      −      1              /            n      )                        |                3                 d  x  =  ∞  .

Does lim_{n to infty}int_a^b f(x)^{p_n(x)}dx=int_a^bf(x)^{p(x)}dx \end{align*} hold in general?

Answered question

Answer & Explanation

New Questions in Calculus 1