Let $\mathcal{B}(\mathbb{R})$ be the D5315: Standard real borel sigma-algebra.
Let $\{ W_t \}_{t \in [0, \infty)}$ be a D3658: Standard real Wiener process.
A D5076: Random real process $B : [0, 1] \to \text{Random}(\Omega \to \mathbb{R})$ is a standard real brownian bridge process if and only if \begin{equation} \forall \, t \in [0, 1] : \forall \, E \in \mathcal{B}(\mathbb{R}) : \mathbb{P}(B_t \in E) = \mathbb{P}(W_t \in E \mid W_1 = 0) \end{equation}