The following question came up in a discussion with my advisor:

Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-algebra of $\mathcal F$. Does there exist an event $U \in \mathcal F$ such that

$U$ is independent of $\mathcal G$, and

$0 < \mathbb P(U) < 1$ ?

The answer is surely yes, but we can't seem to prove it. Our initial approach was to consider the space $L^2(\Omega, \mathcal G)^\perp$ of finite-variance random variables orthogonal to $\mathcal G$ (i.e. $\mathbb E(X|\mathcal G) = 0$ a.s.). Since $\mathcal G$ is a proper sub-$\sigma$-algebra of $\mathcal F$, this space is non-trivial. Choose a non-constant $X \in L^2(\mathcal G)^\perp$ and let $U = \{X < c\}$. For some $c$ the event $U$ has non-trivial probability.

However, orthogonality does not imply independence. For a simple example, take independent random variables $Y \sim \operatorname{Bernoulli}(1,p)$ and $Z \sim N(0,1)$, then set $\mathcal G = \sigma(Y)$ and $X = YZ$. Certainly $X$ is not independent of $\mathcal G$ though one easily can check that $\mathbb E(X|\mathcal G) = 0$ a.s. Our strategy above might find $X$; could we modify our strategy to find the independent $Z$ instead?