Consider the interval \([0,1]\) endowed with Lebesgue measure \(\lambda\) on the Borel \(\sigma\)-algebra \(\mathcal{B}\). Define \(\mathcal{F}_{t}=\sigma\{A: A \subset[0, t], A \in \mathcal{B}\}\). Let \(f\) be an integrable function defined on \([0,1]\), considered as a random variable.

\[\mathbb{E}\left(f \mid \mathcal{F}_{t}\right)(u)=f(u) \mathbb{1}_{\{u \leq t\}}+\mathbb{1}_{\{u>t\}} \frac{1}{1-t} \int_{t}^{1} d x f(x)\]