 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2011/a/1/b/ii/solution

Each half-line

(- \infty, x]

is a closed Borel set, and the empty set is Borel. Thus

σ (π (R)) \subseteq B (R)

Conversely,

(- \infty, b) = ⋃_{n \geq 1} (- \infty, b - 1/ n], (a, b) = (- \infty, b) ∖ (- \infty, a] .

The generated sigma-algebra contains both sets on the right and is closed under their difference, so it contains every open interval. By the given definition of the Borel sigma-algebra, the reverse inclusion follows. Therefore

σ (π (R)) = B (R) .

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