Union of sets of sets I am not sure how one should interpret the union of Sets of sets. Is the union a set with the elements being the elements of the sets inside the Sets, or is it a set with the elements being the Sets?
 A: Perhaps using plain language here can confuse, so
(1) Interpreting Sets of sets literally, Let $\mathscr X$ and  $\mathscr Y$ be "sets of sets", with $\mathscr X$ = {$X_1, X_2, ..$} and $\mathscr Y$ = {$Y_1, Y_2, ..$} where the $X_i$ and $Y_i$ are sets then $\mathscr X \cup \mathscr Y$ = {$X_1, X_2, ..Y_1, Y_2, ..$} and $\mathscr X \cup \mathscr Y$ is a set of sets.
(2) If we are talking about sets of sets generally so that $\mathscr X$ is one such "set of sets" and $\mathscr X$ = {$X_1, X_2, ..$} where each $X_i$ is a set = {$x_{i,1 }, x_{i,2 }, ... $} then you can form the union of the set of sets in $\mathscr X$, notated as $\cup \mathscr X$ = {$x_{1,1 }, x_{1,2 }, ... x_{2,1 }, x_{2,2 }, ...$}  
A: Let $A, B, C, D, E$ be sets.
$A = \{0, 1, 2, \cdots 9\}$
$B = \{0, 2, 4, \cdots, 18\}$
$C = \{1, 3, 5, \cdots, 19\}$
$D = \{10, 11, 12, \cdots 19\}$
$E = \{0, 5, 10, 15, 20\}$
Let $P = \{A, B, E\}$ and $Q = \{C, D\}$.
Then $P\cup Q = \{A, B, C, D, E\}$. 
Here, the elements of $P, Q$ are sets, so their union is a set of sets (a set whose elements are sets). So their union is not a set of all the set elements contained in $A, B, C, D, E$. (I.e., none of the numbers $0$ to $20$ are in $P, \;Q,\;$ OR $P\cup Q$.)

If you want to describe the union of all elements in $A, B, C, D, E$, you would write, e.g., $A \cup B \cup C\cup D\cup E = \{0, 1, 2, 3, \ldots, 18, 19, 20\}$.
A: A set (say $X$) of sets (say $A$) is just a set where all of it's elements are sets themselves. Say we are living in $\mathbb{R}^3$, then if we let $X_1=\{A|m(A)=1\}$, then $X$ is the set of all sets with measure $A$ equal to 1.
If we take $\bigcup_{\alpha\in\mathbb{R}^+}X_\alpha$, then this union of sets of sets, contains every set $X_\alpha$, which itself contains all the sets of measure $\alpha$.
It would be wrong to say that the elements of the union are all the sets with some positive measure $\alpha$, since the $A$'s are all elements of $X_\alpha$'s themselves. The elements of the union are all the sets of sets $X_\alpha$.
Hopefully this example helps :)
A: 
Is the union of a set with the elements being the elements of the sets inside the Sets...

Yes. More precisely, let $F=\{s_1, s_2, s_3, ...\}$ where the $s_i$ are sets. Then $x\in\cup F$ (the union of $F$) if and only if $x$ is an element of at least one of the $s_i$.
Likewise $x\in\cap F$ (the intersection of $F$) if and only if $x$ is an element of every one of the $s_i$.
