Supremum and Infimum

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#1

Now, here's where my problem comes in:

I know that the union of measurable sets is measurable, and if I could just state the above, I'd be fine, but it's asking me to show it, and I have no idea how to start

Does anyone have any idea where I can start to solve this?
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6 years ago
#2
Maybe (probably!) I'm being stupid, but is the first identity actually true?

Take the measurable space as [0,1] with standard measure (though it hardly matters what we choose), and define X_n(x) = 1 - 1/n (so a constant function).
Then , but so .
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