expectation of the weighted sum of random variables

Question

Let $X_{i}(\omega)\geq 0$, for $i=1,\dots,k$, and $Y(\omega)\geq 0$ be the random variables defined on the same probability space $(\Omega, F,P)$, such that $E[X_{i}] \leq E[Y]$ for every $i$. Next, let $A_{i}$ be disjoint elements of the sigma-algebra $F$ such that $\cup_{i}A_{i} = \Omega$. Are the following statements correct: $$ \sum_{i}^{k}E[X_{i}]P[A_{i}] \leq E[Y] $$ and $$ \sum_{i}^{k}E[X_{i}I\{A_{i}\}] \leq E[Y] $$ ?

Answer 1

Since $\sum_i P(A_i) = 1$ and $E(X_i) \leq E(Y)$,

$$\sum_i E(X_i)P(A_i) \leq \sum_i E(Y)P(A_i) = E(Y).$$

The second inequality is false. Let $\Omega = \{1,2\}$, $X_i(i) = 2$ and $X_i(j)=0$ if $i \neq j$, $i,j \in \{1,2\}$. Let $P(i) = 1/2$, $i \in \Omega$. Then, $E(X_i) = 1$, so if $Y = 1$, then $E(X_i) \leq E(Y)$. But with $A_i = \{i\}$,

$$E(X_1I_{A_1}) + E(X_2 I_{A_2}) = 1 + 1 = 2 > 1 = E(Y).$$