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 \}
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s