Why does $P(X=x)=0$ when $x$ is a continuous random variable? Apart from reasoning using integration, in terms of events how do we make sense of this? Why do we say a probability AT any value is zero?
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Does this answer your question? Why is the probability that a continuous random variable takes a specific value zero? – Yalikesifulei Jul 30 '22 at 06:07
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This one might also be valuable How to explain why the probability of a continuous random variable at a specific value is 0? – adrien_vdb Jul 30 '22 at 15:42
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Let $X$ be a real-valued random variable on a probability space $(\Omega,\mathcal F,\mathbb P)$ with distribution $\nu$ - that is, for each $E\in\mathcal F$, we have $\mathbb P(X\in E) = \nu(E)$. By definition, $X$ is continuous when there exists a measurable function $f$ which satisfies $\nu(E) = \int_E f\ \mathsf d\lambda$ where $\lambda$ is Lebesgue measure on $\mathbb R$. Since the Lebesgue measure of a finite set is zero, it follows readily that $$\nu(\{0\})=\mathbb P(X=0) = \int_{\{0\}} f\ \mathsf d\lambda = 0.$$
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