Intuition of $ P( X = a) $ for a continuous random variable? Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual
 measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.

The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the definition of the probability mass function. I am not sure if my question makes sense, but how should I think of $ P( X = a) $ when $X$ is a continuous random variable?
This doesn't need any measure theory.


What's the probability of picking 0.5 from the interval [0,1] following a [insert continuous eg uniform] distribution?
Can you measure 100% precisely 1.1566672 cm with a ruler?
Open Excel/Google Sheets and in two cells simulate values from a continuous distribution eg normal. In a third cell, check whether the values in the two cells are equal. I don't think they will ever be equal.
This doesn't need any measure theory.


What's the probability of picking 0.5 from the interval [0,1] following a [insert continuous eg uniform] distribution?
Can you measure 100% precisely 1.1566672 cm with a ruler?
Open Excel/Google Sheets and in two cells simulate values from a continuous distribution eg normal. In a third cell, check whether the values in the two cells are equal. I don't think they will ever be equal.