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Suppose I want to solve the following problem

$$\max_{f(\cdot)} \int f(x) g(x) d\mu(x)$$

where the maximization is over measurable functions $f:X\to [0,1]$ , $\mu$ 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?

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  • $\begingroup$ If you maximize over all measurable functions you will always get $\infty$ as long as $\int g d\mu$ is not zero. $\endgroup$ Aug 9, 2019 at 23:40
  • $\begingroup$ Thanks. I forgot to write that $f$ goes to $[0,1]$. Fixed it. $\endgroup$
    – Condor5
    Aug 9, 2019 at 23:42
  • $\begingroup$ I think you still need some condition on $g$. For example, take $\mu$ to be the Lebesgue measure restricted to the interval (0, 1] and $g(x)=1/x$ for $x > 0$. Then $\int g d\mu = \int_0^1 g(x) dx = +\infty$. Now take $f(x)=1$. $\endgroup$ Aug 10, 2019 at 2:19
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    $\begingroup$ Maybe $g$ integrable is missing? Then $\int f(x) g(x) d\mu(x) = \int f(x) g^+(x) d\mu(x) - \int f(x) g^-(x) d\mu(x) \leq \int g^+(x) d\mu(x)$. And the maximum is therefore reached for $f = 1_{X^+}$, where $X^+ = \{x \in X \ s.t. \ g(x) > 0 \}$ $\endgroup$
    – user605486
    Aug 10, 2019 at 2:31
  • $\begingroup$ I'm looking for conditions for there not be a solution. Can anyone think of an example? $\endgroup$
    – Condor5
    Aug 11, 2019 at 19:08

1 Answer 1

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If $g$ is not integrable the problem can be unbounded: Let $\mu$ 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 $\int f g d\mu = \int |x| d\mu(x) = \infty$.

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