Is the event $\{\max\{X_1,X_2\}=X_2\}$ measurable with respect to $\sigma(\max\{X_1,X_2\})$?

It is not true. Let $X_1$ be a Bernoulli(1/2) random variable and set $X_2=1-X_1$. Then $\max(X_1,X_2)$ is the constant random variable 1. The generated $\sigma$-algebra $\sigma(\max\{X_1,X_2\})$ is trivial, but the event $\{\max\{X_1,X_2\}=X_2\}$ is not.

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Probability Theory

Measure Theory

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