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I have the following problem. I need to consider all the conditions in which the following integral may be equal to zero:

$$\int_\Omega [p\phi-\lambda(\omega)]f(\omega)\iota(\omega)d\omega$$

Where $p>0$ is a constant. $f$ is a probability density function and $\iota$ is an indicator function (i.e I am truncating the density). $\lambda(\cdot)$ is an increasing function in $\omega$ and $\phi$ is a random variable, so I am solving this integral for any possible realization of $\phi$. Of course there is a trivial solution when $p\phi=\lambda(\omega*)$ for some $\omega*$. But there exist any other case when this is zero which I am not considering?

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  • The expectation $\mathbb{E}\left[\iota(p\phi-\lambda)\right]$ could be zero depending on the value of $p\phi-\lambda$ on all $\omega$.
  • If there is no $\omega$ for which $\iota(\omega)=1$, again you end up with a zero.
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