I have non-negative function g ( y , x ) that is define using non-negative f ( y , x ) in the following way: g ( y , x ) = ∫ 0 x f ( t , y ) d t I am trying to maximize max y ∈ S g ( y , x ). Using Fatou's lemma we have that max y ∈ S g ( y , x ) = ∫ 0 x f ( t , y ) d t ≤ ∫ 0 x max y ∈ S f ( t , y ) d t I also have that max y ∈ S f ( t , y ) ≤ h ( t ) where ∫ 0 x h ( t ) &lt; ∞My question when does the last inequality hold with equally?What do I have to assume about f ( y , x ) and g ( x , y ) for equality to hold?Can I apply dominated convergence theorem here?

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I have non-negative function   g  (  y  ,  x  ) that is define using non-negative   f  (  y  ,  x  ) in the following way:                    g        (        y        ,        x        )        =                  ∫                      0                    x                f        (        t        ,        y        )        d        t            I am trying to maximize       max          y      ∈      S        g  (  y  ,  x  ). Using Fatou&#039;s lemma we have that                              max                      y            ∈            S                          g        (        y        ,        x        )        =                  ∫                      0                    x                f        (        t        ,        y        )        d        t        ≤                  ∫                      0                    x                          max                      y            ∈            S                          f        (        t        ,        y        )        d        t            I also have that       max          y      ∈      S        f  (  t  ,  y  )  ≤  h  (  t  ) where       ∫    0    x    h  (  t  )  &amp;lt;  ∞My question when does the last inequality hold with equally?What do I have to assume about   f  (  y  ,  x  ) and   g  (  x  ,  y  ) for equality to hold?Can I apply dominated convergence theorem here?

Narissiyks · Accepted Answer

Consider       y    n   such that   g  (      y    n    ,  x  )  →      max          y      ∈      S        g  (  y  ,  x  ).Now you obtain under the assumption that       sup    y    f  (  t  ,  y  ) is well behaved and bounded      max          y      ∈      S        g  (  y  ,  x  )  =      lim    n    g  (      y    n    ,  x  )  =      lim    n        ∫    0    x    f  (  t  ,      y    n    )    d  t  =      ∫    0    x        lim    n    f  (  t  ,      y    n    )    d  tTherefore the equality                              max                      y            ∈            S                          g        (        y        ,        x        )        =≤                  ∫                      0                    x                          max                      y            ∈            S                          f        (        t        ,        y        )        d        t            holds if       lim    n    f  (  t  ,      y    n    )  =      sup          y      ∈      S        f  (  t  ,  y  ).Under the additional hypothesis that   S is compact. Then       y    n    →      y    ∗   follows without loss of generality and equality will hold if there is a       y    ∗   such that  f  (  t  ,      y    ∗    )  =      max          y      ∈      S        f  (  t  ,  y  )

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