What does the "$+$" typically denote when summing random variables? Let $X_1$ and $X_2$ be two random variables.
When in the literature (for example, in the context of the law of large numbers) one sees statements along the lines of
$$
X = X_1 + X_2
$$
...does the "$+$" simply denote normal function addition, as in
$$
X(k) = X_1(k) + X_2(k)
$$
for all $k$ the domain of $X$, $X_1$, and $X_2$?  Or does this $+$ denote something else?
 A: I completely understand your frustration with the ambiguous and inconsistent notation often used in statistics, even in reputable sources. It is one of the main reasons why I keep having trouble studying it, because I can't help but constantly ask questions like: "Almost surely with respect to what measure? What kind of convergence is implied?..." and the answers are in my opinion not always obvious from the context. Mathematical statisticians, please get it together.
That being out of the way: remember that random variables are essentially just measurable functions $(\Omega, \mathcal{M}, \mathcal{P}) \to (\mathbb{R}^n, \mathcal{R}^{\otimes n})$ where $(\Omega, \mathcal{M}, \mathcal{P})$ is some probabiilty space, $n \in \mathbb{N}$ and $\mathcal{R}$ is the Borel $\sigma$-algebra on $\mathbb{R}$. If you have two functions $X_1, X_2 : (\Omega, \mathcal{M}, \mathcal{P}) \to (\mathbb{R}^n, \mathcal{R}^{\otimes n})$ (same probability space and same $n \in \mathbb{N}$) then one can use them to define new random variables just by using pointwise operations, e.g.
$$
X_1 + X_2 : (\Omega, \mathcal{M}, \mathcal{P}) \to (\mathbb{R}^n, \mathcal{R}^{\otimes n}) : x \mapsto X_1(x) + X_2(x)
$$
I believe that this is by far the most common interpretation of $X_1 + X_2$. One can show that this function is still measurable. In fact, since every continuous function $f : \mathbb{R}^n \to \mathbb{R}^m$ is measurable and since compositions of measurable functions are again measurable, every continuous function $f$ in $(x_1, x_2)$ will define a new random variable
$$
f(X_1, X_2) : (\Omega, \mathcal{M}, \mathcal{P}) \to (\mathbb{R}^m, \mathcal{R}^{\otimes m}) : x \mapsto f(X_1(x), X_2(x)).
$$
The case $f(x_1, x_2) = x_1 + x_2$ covers your problem. The distribution $\mathcal{P}_{f(X_1, X_2)}$ of $f(X_1, X_2)$ is always defined: for any set $A \in \mathcal{R}^{\otimes m}$ we have
$$
(\mathcal{P}_{f(X_1, X_2)})(A) = \mathcal{P}(\lbrace x \in \Omega \vert f(X_1(x), X_2(x)) \in A \rbrace)
$$
but it might not be trivial to find a pdf, as was mentioned in user1952009's comment.
The ambiguity you talk about is often not in the "$+$" sign, but in the "$=$" sign. Indeed, it is sometimes used to denote "has the same distribution as". Two stochastic variables $X_1, X_2: (\Omega, \mathcal{M}, \mathcal{P}) \to (\mathbb{R}^n, \mathcal{R}^{\otimes n})$ are said to have the same distribution if for every $A \in \mathcal{R}^{\otimes n}$
$$
(\mathcal{P}_{X_1})(A) = (\mathcal{P}_{X_2})(A)
$$
where, as before, we define
$$
(\mathcal{P}_{X})(A) = \mathcal{P}(\lbrace x \in \Omega \vert X_1(x) \in A \rbrace)$$
Obviously if two variables are equal, then they have the same distribution, but certainly not the other way around. See this question for some examples on this phenomenon.
A: Remember that $X_{1}:\Omega\to [0,1]$ and $X=X_1+X_2$ means, for each $\omega\in \Omega$, that $X(\omega)$ is the value of $X_1(\omega)+X_2(\omega)$. Note that is well defined, because are real numbers.
