In common continuous distributions there is usually a way to define a probability density for each state

However, many applications of probability require what are known as continuous distributions (such as the uniform/rectangular, normal, and beta distributions), and thus require a restriction to countable additivity. In a continuous distribution, there are uncountably many states, usually named by real numbers. Each individual state has probability 0, even though events containing uncountably many states often have non-zero probability. (This violates full additivity.) However, in the common cont