Consider a random vector $\eta\equiv (\eta_1,...,\eta_J)$ with support $\mathbb{R}^J$.
Let $\mathcal{J}\equiv \{1,...,J\}$ and let us partition $\mathcal{J}$ into two sets, $\mathcal{J}_1$ and $\mathcal{J}_2$, with cardinalities denoted by $|\mathcal{J}_1|$ and $|\mathcal{J}_2|$.
Consider some known parameters $\alpha_1,...,\alpha_J, \beta$.
Let $\bar{\eta}$ be a realisation of $\eta$ (i.e., a $J\times 1$ vector of real numbers).
Assume that $$ \alpha_j-\beta -\bar{\eta}_j\geq 0 \hspace{1cm}\forall j \in \mathcal{J}_1 $$
Claim to show: $$ E(\alpha_j -\beta-\eta_j| j\in \mathcal{J}_1)\geq 0 $$
I am struggling to understand what the claim means and how it is shown. Could you help? There could be hidden conditions to add, please suggest them if you think they are implicit.
My thoughts:
Fact 0: How is the conditional expectation $E(\alpha_j -\beta-\eta_j| j\in \mathcal{J}_1)$ defined? The only way I can think of is by considering the sets $\mathcal{J}_1$ and $\mathcal{J}_2$ as "random". Then, we define a random variable $Q_j$ taking value $1$ if $j\in \mathcal{J}_1$ and $0$ otherwise. Hence, $$ E(\alpha_j -\beta-\eta_j| j\in \mathcal{J}_1)=E(\alpha_j -\beta-\eta_j| Q_j=1)\equiv \sum_{\bar{\eta}_j\in \mathbb{R}} \Big[\alpha_j -\beta-\bar{\eta}_j\Big] Pr(\eta_j=\bar{\eta}_j| Q_j=1) $$
Fact 1: $$ \overbrace{\alpha_j-\beta -\bar{\eta}_j}^{\text{This is just a number!}}\geq 0 \hspace{1cm}\forall j \in \mathcal{J}_1 $$ $$ \Downarrow $$ $$ \overbrace{\frac{1}{|\mathcal{J}_1|}\sum_{j=1}^{\mathcal{J}_1} \Big[ \alpha_j-\beta -\bar{\eta}_j\Big]}^{\text{This is just a number!}}\geq 0 $$
Fact 2: $$ \text{ If $|\mathcal{J}_1|$ large and $\{\eta_j\}_{j\in \mathcal{J}_1}$ i.i.d., then }\\\frac{1}{|\mathcal{J}_1|}\sum_{j=1}^{\mathcal{J}_1} \Big[ \alpha_j-\beta -\eta_j\Big]\to_{a.s.} E(\alpha_j -\beta-\eta_j| Q_j=1) \\ \text{ by law of large numbers} $$ Therefore, $\frac{1}{|\mathcal{J}_1|}\sum_{j=1}^{\mathcal{J}_1} \Big[ \alpha_j-\beta -\bar{\eta}_j\Big]$ is a good approximation of $E(\alpha_j -\beta-\eta_j| Q_j=1)$ when $J$ is very large. Hence, we can conclude that the claim should hold.
Is this the correct interpretation? It does not actually convince me what I wrote because if $\mathcal{J}_1$ is "random", then also $|\mathcal{J}_1|$ is random...