Indicator function and expectation Consider a random variable $X$ uniformly distributed over $[0,1]$ and the indicator function $\mathbb{1}_{X \geq \tilde{x}} $ equal to one if $X \geq \tilde{x}$ and zero otherwise. We know that $$\begin{align}
\mathbb{E} (\mathbf{1}_{X \geq \tilde{x}})
 & = \mathbb P(X\geq \tilde x)
\\[1ex] & = 1 - F_X(\tilde{x})
\end{align}$$ 
My question is if I draw an $X$ strictly below $\tilde{x}$ is the above probability equal to zero? 
Thank you
 A: I will answer your question with a question:  suppose you have a fair six-sided die and you roll it.  You are interested in the probability of observing a value of at least $3$; e.g., if $X$ is a random variable that takes on values $X \in \{1, 2, 3, 4, 5, 6\}$, each with equal probability, then you are interested in the probability $\Pr[X \ge 3]$.  Now, suppose you observed $X = 1$:  you rolled a $1$.  Does that now somehow magically mean that $\Pr[X \ge 3] = 0$?  Does that mean the die is not actually fair?  Of course not.
Your confusion stems from not making a clear distinction between probabilities, events, and outcomes.  Loosely (non-mathematically) speaking, outcomes are the results of some underlying random process; events are collections of outcomes; and probabilities are values we assign to events that capture in some sense how likely it is we are to observe that event.  But if an event is not observed/realized, that doesn't mean the probability of that event is zero.  If I flip a fair coin and I get tails, it doesn't mean the probability of heads is zero.
Now, if you define an indicator function as you have done in your example, the value of that function is zero if the event attached to that indicator doesn't occur.  All that an indicator function does is give you a Bernoulli random variable that corresponds to whether an event of interest occurs.  So with the die example, I could create the indicator function $$\mathbb 1_{X \ge 3} = \begin{cases} 1, & X \in \{3, 4, 5, 6\} \\ 0, & X \in \{1, 2\}. \end{cases}$$  All this does is define a random variable that is $1$ if you roll a $3, 4, 5$ or $6$, and $0$ if you roll a $1$ or $2$.  You could also create another indicator such as $$\mathbb 1_{X \in \{1,3,5\}} = \begin{cases} 1, & X \in \{1,3,5\} \\ 0, & X \in \{2,4,6\} \end{cases}$$ which is $1$ if $X$ is odd and $0$ if $X$ is even.
