## MA5161. HW 1 due Wednesday 1/25

Exercise 1. Show that the  $\sigma$-algebra $\mathcal{T}(\mathbb{R}_{\ge 0},\mathbb{R}^d)$ is also generated by the following families:

• $\{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d ), f(t_1) \in B_1,...,f(t_n) \in B_n \}$where $t_1,...,t_n \in \mathbb{R}_{\ge 0}$ and where $B_1,...,B_n$ are Borel sets in $\mathbb{R}^d$.
• $\{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), (f(t_1),...,f(t_n)) \in B \}$where $t_1,...,t_n \in \mathbb{R}_{\ge 0}$ and where $B$ is a Borel set in $(\mathbb{R}^{d})^{\otimes n}$.

Exercise 2.  Show that the following sets are in $\mathcal{B} ([0,1],\mathbb{R})$:

• $\{ f \in \mathcal{C}([0,1], \mathbb{R}), \sup_{t\in [0,1]} f(t) <1 \}$
• $\{ f \in \mathcal{C}([0,1], \mathbb{R}), \exists t\in [0,1] f(t) =0 \}$
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