Consider a probability space $(\Omega,\mathcal{F},P)$ and a linear space $\mathcal{M}\subset \mathcal{L}^0(\Omega,\mathcal{F},P)$, where $\mathcal{L}^0(\Omega,\mathcal{F},P)$ denotes the set of all random variables on the probability space that are almost surely finite.
I'm new to probability theory, in fact, I'm not that interested in the theory itself, but I'm reading an economics textbook and want to understand the above.
Can you help me understand, in simple terms, or "visualize" what a linear space is like (and more specifically, the space $\mathcal{L}^0$)?
Almost surely finite means that the random variables are finite with probability 1, correct? What would be an example of this -- a normally distributed random variable?