Let $(\Omega,\mathcal A,\mu)$ be a probability space and $$S:=\left\{w\in\mathcal L^2(\mu)\times \mathcal L^2(\mu):w_1+w_2=1\right\}.$$
What is the (topological) boundary of $S$?
I'm a bit lost. Is there an easy approach to the problem?
2026-04-01 02:01:54.1775008914
How can we determine $\partial\left\{w\in\mathcal L^2(\mu)\times \mathcal L^2(\mu):w_1+w_2=1\right\}$?
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I would interpret $w_1+w_2=1$ as equality of equivalence classes, i.e. equality almost everywhere.
Hint: this is a closed set with no interior so it is its own boundary. [If $w_1+w_2=1$ you can find $w_3$ as close to $w_2$ as possible with $w_1+w_3 \neq 1$. So $(w_1,w_3)$ is close to $(w_1,w_2)$ but $w_1+w_3 \neq 1$.]