Thomas E.
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 Sep 11 comment $L_p$ norm not subadditive for $050\%$ probability. @sTEAK. If you take one from each jar then the probability of getting a red is $1$. Aug 16 comment Is it possible to have $D=\Bbb P$ @t.b. Yeah, the countability is important :) Baire implies that any countable completely metrizable topological space has an isolated point. If not, you could write the whole space as a countable union of nowhere dense closed sets (singletons), making the whole space to have an empty interior, which is a contradiction. Aug 16 comment Is it possible to have $D=\Bbb P$ @t.b. Yeah, I meant an isolated point. And true, that also does the job. Thanks for providing an alternative way of seeing it. Aug 16 revised Is it possible to have $D=\Bbb P$ added 462 characters in body Aug 16 answered Is it possible to have $D=\Bbb P$ Aug 14 comment Is the set of irrationals separable as a subspace of the real line? @CameronBuie. To add on Asaf's comment, it's worth noting that the word 'metric space' plays a key role there. Metrizability is hereditary and so is second countability. Since in a metric space separability (also Lindelöf) is equivalent with second countability, then every subspace of a separable metric space is separable (and Lindelöf, for that matter). Aug 13 comment Baire Category Theorem @AsafKaragila. True. The ZF part was so well hidden in the link that I totally missed it. Aug 13 comment Baire Category Theorem By the way, this works also for non-separable spaces. And you could try to show the complement of this statement which says that a countable intersection of open dense sets is dense. It's quite simple, too. You begin with an arbitrary open ball and start choosing closed balls inside it inductively so that the radius goes to zero and the $n$:th step ball is a subset of $n$ first members of the dense open sets. The intersection of these closed balls is a singleton by completeness: it belongs to the intersection of the open dense sets and our original ball, proving the claim. Aug 13 comment $L^p$-norm of a non-negative measurable function Hint: Assume first that $f$ is a simple function and show that the equation holds. Then for your non-negative measurable $f$ choose a nondecreasing sequence of simple functions converging point-wise to $f$ and use monotone convergence theorem.