I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure)
It is hard to google this, too. What does such a random variable mean? If it helps, it was defined given a probability space $([0,1],B(0,1),L)$ where $B(0,1)$ contains Borel sets intersecting $[0,1]$ and $L$ is the Lebesgue measure.
Is there a general way to google specific notations that I might miss? I have skimmed through the wiki articles of Borel sets and the Lebesgue measure without success.
For a set $\mathbf{A}$, $\mathbf{1}_\mathbf{A}$ often designates the characteristic function of $\mathbf{A}$, that is, the function defined by $\mathbf{1}_\mathbf{A}(x) = 1 \text { if } x \in \mathbf{A}$ and $\mathbf{1}_\mathbf{A}(x) = 0$ otherwise.
When $\mathbf{A}$ is measurable, so is $\mathbf{1}_\mathbf{A}$ and $\int \mathbf{1}_\mathbf{A}d\mu = \mu(\mathbf{A})$
In your case, $\mathbf{A} = [0, \frac{1}{2}]$ and $\mu$ is the Lebesgue measure, so $\mathbf{1}_{[0,\frac{1}{2}]}$ is a random variable of expectation $\frac{1}{2}$.