I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as follows.

Consider a random variable $X$ supported on $\mathbb{R}^{d}$ with distribution $P_X$. Assume we have constructed a cover of $\mathbb{R}^{d}$ denoted as $\{\Lambda(\omega):\omega\in\Omega\}$, where $\Omega$ is a finite set. We then make $\Omega$ into some probability space and introduce an extra r.v. $\sigma$ that takes value in $\Omega$.

Now since we have (or not, why?)

$$ \mathbb{E}_{\sigma}\int_{\Lambda(\sigma)}XX^{T}dP_{X} = \mathbb{E}_{X}[XX^{T}] $$,

there should exist some constant $\lambda >0$, such that the following operator inequality is true with probability (in the order of positive semi-definiteness)

$$ P_{X}(\Lambda(\sigma))\mathbb{E}_{X}[XX^{T}] - \lambda{I} \preceq \int_{\Lambda(\sigma)}XX^{T}dP_{X}\preceq P_{X}(\Lambda(\sigma))\mathbb{E}_{X}[XX^{T}] + \lambda{I} $$

From my perspective, the existence may be easy as we could somehow add some additional assumptions on $P_X$ to assert the "boundedness" of $XX^{T}$. Thus we could take $\lambda$ larger than the upper bound.

Could anyone provide some reference or hints relevant to this problem? That will help a lot.