Suppose I want to solve the following problem max f ( ⋅ ) ∫ f ( x ) g ( x ) d μ ( x )where the maximization is over measurable functions f : X → [ 0 , 1 ], μ is a finite measure and g is measurable.Can someone come up with an example such that the problem does NOT have a solution? I imagine one can try to maximize point by point and obtain something that is not measurable and then the problem would not have a solution?

Question

Suppose I want to solve the following problem      max          f      (      ⋅      )        ∫  f  (  x  )  g  (  x  )  d  μ  (  x  )where the maximization is over measurable functions   f  :  X  →  [  0  ,  1  ],   μ is a finite measure and   g is measurable.Can someone come up with an example such that the problem does NOT have a solution? I imagine one can try to maximize point by point and obtain something that is not measurable and then the problem would not have a solution?

wasipewelr · Accepted Answer

If   g is not integrable the problem can be unbounded: Let   μ be the Cauchy distribution on the real line. Let   g  (  x  )  =      |    x      |  . Then, we can take   f  (  x  )  =  1 for all   x and we have   ∫  f  g  d  μ  =  ∫      |    x      |    d  μ  (  x  )  =  ∞.

Suppose I want to solve the following problem <munder> <mo movablelimits="true" form="pr

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