How can I show using the monotone convergence theorem that the Lebesgue integral ∫ 0 ∞ sin ⁡ ( x ) x d xdoes not exist?

Question

How can I show using the monotone convergence theorem that the Lebesgue integral      ∫          0        ∞              sin      ⁡      (      x      )        x      d  xdoes not exist?

escampetaq5 · Accepted Answer

Recall that   f is (Lebesgue) integrable if and only if       |    f      |   is Lebesgue integrable. Now, we may write      f    n    (  x  )  =      ∑          k      =      0        n                      |            sin      ⁡      x              |              x        χ          [      k      π      ,      (      k      +      1      )      π      ]      so that       f          n        ≤      f          n      +      1       and       f    n    →  f pointwise, where   f  (  x  )  =                    |            sin      ⁡      x              |              x  . We may then estimate the integral of each term in the sum via:      2          (      k      +      1      )      π        =      ∫          k      π              (      k      +      1      )      π                          |            sin      ⁡      x              |                    (      k      +      1      )      π        d  x  ≤      ∫          k      π              (      k      +      1      )      π                          |            sin      ⁡      x              |              x    d  xSo that by the monotone convergence theorem,                              ∫          0          ∞                                                    |                        sin            ⁡            x                          |                                x                d        x                            =                  lim                      n            →            ∞                                    ∫          0          ∞                          f          n                d        x                                          ≥                  ∑                      k            =            0                    ∞                          2                      (            k            +            1            )            π                                                            =        ∞            Which implies that the Lebesgue integral of             sin      ⁡      x        x   does not exist.

How can I show using the monotone convergence theorem that the Lebesgue integral <msubsup> &

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