Let $E$ be a topological space and $\mathcal P (E)$ the space of all Borel probability measures on $E$. We endow $\mathcal P (E)$ with the topology of weak convergence. Could you have a check on my attempt of below theorem?
Theorem: $\mathcal P (E)$ is connected.
Proof: Fix $\mu, \nu \in \mathcal P (E)$ and Consider the map $$ \varphi: [0, 1] \to \mathcal P(E), t \mapsto t \mu + (1-t) \nu. $$
Then $\varphi (1) = \mu$ and $\varphi (0) = \nu$. Let's prove that $\varphi$ is continuous. Let $(t_n) \subset [0, 1]$ such that $t_n \to t \in [0, 1]$. Let $f:E \to \mathbb R$ be bounded continuous. Then $$ \begin{align} \int f \mathrm d [t_n \mu + (1-t_n) \nu] &= t_n \int f \mathrm d \mu + (1-t_n) \int f \mathrm d \nu \\ &\xrightarrow{n \to \infty} t \int f \mathrm d \mu + (1-t) \int f \mathrm d \nu \\ &= \int f \mathrm d [t \mu + (1-t) \nu]. \end{align} $$
Hence $t_n \mu + (1-t_n) \nu \rightharpoonup t \mu + (1-t) \nu$. It follows that $\mathcal P (E)$ is path-connected and thus connected.