"expectation of sum is sum of expectation", is this claim true? if yes, how to justify this claim? this post is saying


  linearity of expectation gives following equation
  $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$


per wiki, Linearity of Expected_value is saying

$$
{\displaystyle {\begin{aligned}\operatorname {E} [X+Y]&amp;=\operatorname {E} [X]+\operatorname {E} [Y],\\[6pt]\operatorname {E} [aX]&amp;=a\operatorname {E} [X],\end{aligned}}}$$

where $X$ and $Y$ are arbitrary random variables, and $a$ is a constant.

is not saying some rules like "expectation of sum is sum of expectation"

this post is saying 


  expectation of sum of something = sum of expectation of something even
  when the items involved are not independent.


how to justify this claim?
When $E[X_1], E[X_2], \dots, E[X_n]$ exist
$${\displaystyle {\begin{aligned}
E [\sum_{i=1}^n X_i] &amp;= E[X_1+X_2+\dotsb+X_n] \\
&amp;= E[X_1]+E[X_2]+\dotsb+ E[X_n] \\
&amp;= \sum_{i=1}^n E [X_i]
\end{aligned}}}$$
When $E[X_1], E[X_2], \dots, E[X_n]$ exist
$${\displaystyle {\begin{aligned}
E [\sum_{i=1}^n X_i] &amp;= E[X_1+X_2+\dotsb+X_n] \\
&amp;= E[X_1]+E[X_2]+\dotsb+ E[X_n] \\
&amp;= \sum_{i=1}^n E [X_i]
\end{aligned}}}$$