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Let $ (\Omega, \Sigma, \mu) $ be a measure space and let $f$ be a real valued function on $\Omega $ such that $\mu (x :f(x)<t) $ is finite for all t $\in \mathbb{R}$. Let the number $G>0$ be and given and defined a class of measurable functions $\Omega$ by $C= (g: 0\le g(x) \le 1 $ for all $x$ and $ \int g(x)\mu(dx)=G) $

Then the minimization problem I= $\inf_{g \in C} \int f(x)g(x) \mu(dx)$ is solved by $g(x)= \chi_{(f<s)}(x) + c\chi_{(f=s)}(x)$ where $s=\sup(t: \mu((x :f(x)<t)) \le G) $ and $c\mu ((x :f(x)=s))=G-\mu ((x :f(x)<s))$

My question is, how can one show that g(x) given above is the minimizer.


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I think you have made a typo in your question.The $s$ should be $sup\{t:μ\{x:f(x)<t\}\le G\}$ and $μ(x:f(x)<t)$ is finite for all $t$. – Ben Nov 24 '12 at 16:04
you are right, thanks Ben – jake Nov 24 '12 at 17:30
up vote 6 down vote accepted

Let $h$ be any other member of $C$. To show that $g$ is a minimizer, you need to establish that $I(g) \leq I(h)$, which is equivalent to showing $$ \int f(g-h) \leq 0. $$

To show this, split the range of integration into the sub-level, sup-level, and level sets of $f$; i.e., $\{f<s\}$, $\{f>s\}$, and $\{f=s\}$:

\begin{align} \int f (g-h) &= \int\limits_{\{f<s\}} f (g-h) + \int\limits_{\{f>s\}} f (g - h) + \int\limits_{\{f=s\}} f (g - h) \\ &\leq s \int\limits_{\{f<s\}} (g-h) - \int\limits_{\{f > s\}} f h + \int\limits_{\{f=s\}} s (g-h) \\ &\leq s \int\limits_{\{f<s\}} (g-h) - s \int\limits_{\{f > s\}} h + s \int\limits_{\{f=s\}} (g-h) \\ & \leq s \left( \int\limits_{\{f<s\}} (g-h) + \int\limits_{\{f>s\}} (g-h) + \int\limits_{\{f=s\}} (g-h) \right) \\ & = s \int(g - h) = s (G -G) = 0. \end{align}

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thanks rick, 15char – jake Nov 28 '12 at 3:44

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