I would like to verify my solution for the following problem:Let g : ( 0 , ∞ ) → R be a nonzero, continuous function and define G : [ 0 , 1 ] × [ 1 , ∞ ) → R by G ( x , y ) = g ( x y ) .Show that G is not in L 1 ( [ 0 , 1 ] × [ 1 , ∞ ) ).Solution: ∫ 1 ∞ ∫ 0 1 | G ( x , y ) | d x d y = ∫ 1 ∞ ∫ 0 1 | g ( x y ) | d x d y = ∫ 1 ∞ ∫ 0 y | g ( t ) | y d t d y ≥ ∫ 1 ∞ 1 y inf t ∈ ( 0 , y ] | g ( t ) | d y ≥ inf t ∈ ( 0 , ∞ ) | g ( t ) | ∫ 1 ∞ 1 y d y = ∞I'm a little concerned about taking the infimum over the non compact set (0,y] (g is not defined at 0).Is this correct? And if so, how do I address my concern above? Thanks for your time.

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I would like to verify my solution for the following problem:Let   g  :  (  0  ,  ∞  )  →      R   be a nonzero, continuous function and define   G  :  [  0  ,  1  ]  ×  [  1  ,  ∞  )  →      R   by  G  (  x  ,  y  )  =  g  (  x  y  )  .Show that   G is not in       L    1    (  [  0  ,  1  ]  ×  [  1  ,  ∞  )  ).Solution:      ∫    1    ∞        ∫    0    1        |    G  (  x  ,  y  )      |    d  x  d  y  =      ∫    1    ∞        ∫    0    1        |    g  (  x  y  )      |    d  x  d  y  =      ∫    1    ∞        ∫    0    y                                                                |                    g          (          t          )                      |                                                                            y                      d  t  d  y  ≥      ∫    1    ∞                                                    1                                                                  y                          inf          t      ∈      (      0      ,      y      ]            |    g  (  t  )      |      d  y  ≥      inf          t      ∈      (      0      ,      ∞      )            |    g  (  t  )      |        ∫    1    ∞                                                    1                                                                  y                      d  y  =  ∞I&#039;m a little concerned about taking the infimum over the non compact set (0,y] (g is not defined at 0).Is this correct? And if so, how do I address my concern above? Thanks for your time.

Erik Rivera · Accepted Answer

Choose   T such that       ∫    0          T            |    g  (  t  )      |    d  t  &amp;gt;  0. Note that       ∫    1          ∞            ∫    0          y                          |            g      (      t      )              |              y    d  t  d  y  ≥      ∫    T          ∞            ∫    0          y                          |            g      (      t      )              |              y    d  t  d  y. Now use the fact that       ∫    0          y            |    g  (  t  )      |    d  t  ≥      ∫    0          T            |    g  (  t  )      |    d  t for   y  &amp;gt;  T. Can you finish?

I would like to verify my solution for the following problem: Let g : ( 0 , <mi

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