I will appreciate any help.Consider the following function: g ( x ) = { 1 x , if x ∈ ( 1 , ∞ ) 0 , if x ∈ ( − ∞ , 1 ] Determine the set { p ∈ [ 0 , ∞ ] : g ∈ L p ( λ ) }. I know that the set is equal to ( 1 , ∞ ] . But I am not sure how to calculate the cases p = 0 and p = ∞.In my textbook we have the following definitions for p = 0 and p = ∞: L ∞ ( μ ) = { f ∈ M ( E ) : ∃ R &gt; 0 : | f | ≤ R μ -almost everywhere }and L 0 ( μ ) = { f ∈ M ( E ) : lim t → ∞ μ ( { | f | ≥ t } ) = 0 }where M ( E ) = { f : X → R : f is E - B ( R ) - measurable }.

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I will appreciate any help.Consider the following function:  g      (    x    )    =      {                                        1            x                    ,                          if                     x          ∈                      (            1            ,            ∞            )                                                0          ,                          if                     x          ∈                      (            −            ∞            ,            1            ]                                  Determine the set   {  p  ∈      [    0    ,    ∞    ]    :  g  ∈            L        p        (    λ    )    }. I know that the set is equal to       (    1    ,    ∞    ]  . But I am not sure how to calculate the cases   p  =  0 and   p  =  ∞.In my textbook we have the following definitions for   p  =  0 and   p  =  ∞:            L        ∞        (    μ    )    =  {  f  ∈      M        (          E        )    :  ∃  R  &amp;gt;  0  :      |    f      |    ≤  R        μ  -almost everywhere  }and            L        0        (    μ    )    =  {  f  ∈      M        (          E        )    :      lim          t      →      ∞        μ      (    {          |        f          |        ≥    t    }    )    =  0  }where       M        (          E        )    =  {  f  :  X  →      R    :  f        is         E    -      B        (          R        )    - measurable  }.

Kaydence Washington · Accepted Answer

Hints:      |    g  (  x  )      |    ≤  1      |    1      /    x      |    ≥  t    ⟺        |    x      |    &amp;lt;  1      /    t ... what is the size of the set   {  x  ∈  (  1  ,  ∞  )  :      |    x      |    &amp;lt;  1      /    t  } ?

I will appreciate any help. Consider the following function: g ( x ) </

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