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Equivalent conditions for the integral of a bounded function to be bounded
Let $K \in \mathbb{R}$ be a constant such that $|f(x)| \leq K$ for all $x \in \mathbb{R}$. Let $g:\mathbb{R} \to \mathbb{R}$ be such that for every $x \in \mathbb{R}$ we have $g(x) > 0$ and $g(x) 0$.
Under these conditions, do we have the following integral identity?
\begin{align} \int_{ -\infty}^\infty \bigl| g(x)h(x) \bigr| \, \mathrm{d}x &= K|h|\int_{ -\infty}^\infty \bigl| g(x) \bigr| \, \mathrm{d}x \\ &= K\int_{ -\infty}^\infty |g(x)| \, \mathrm{d}x. \end{align}
Obviously for $h(x) = 1$, we get $\int_{ -\infty}^\infty \bigl| g(x) \bigr|
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