Show that the Borel algebra B is also generated by the collection of all half-openintervals (a, b] = {x ∈ R | a < x ≤ b}. Also show that B is generated by thecollection of all half-rays {x ∈ R | x > a}, a ∈ R.

To show that the Borel algebra B is generated by the collection of all half-open intervals (a, b] = {x ∈ R | a < x ≤ b}, we need to show that any open interval (a, b) can be expressed as a countable union of half-open intervals.

Step 1: Express an open interval as a countable union of half-open intervals.

Consider an open interval (a, b). We can express this as a countable union of half-open intervals as follows:

(a, b) = ⋃_{n=1}^{∞} (a + 1/n, b]

This is because for any x in (a, b), there exists an n such that x is in (a + 1/n, b].

Step 2: Show that the collection of all half-open intervals generates the Borel algebra.

The Borel algebra is the smallest σ-algebra containing all open intervals. Since we have shown that any open interval can be expressed as a countable union of half-open intervals, it follows that the Borel algebra can be generated by the collection of all half-open intervals.

To show that B is generated by the collection of all half-rays {x ∈ R | x > a}, a ∈ R, we need to show that any open interval (a, b) can be expressed as a countable union or intersection of half-rays.

Step 3: Express an open interval as a countable union or intersection of half-rays.

Consider an open interval (a, b). We can express this as the intersection of two half-rays as follows:

(a, b) = {x ∈ R | x > a} ∩ {x ∈ R | x < b}

This is because for any x in (a, b), x is greater than a and less than b.

Step 4: Show that the collection of all half-rays generates the Borel algebra.

The Borel algebra is the smallest σ-algebra containing all open intervals. Since we have shown that any open interval can be expressed as a countable union or intersection of half-rays, it follows that the Borel algebra can be generated by the collection of all half-rays.