Continuous as follows: For n 1, 2,€¦, letX_n,Xbe r.v.s defined on the measure space (W, A, μ), and suppose that

Then, by means of concrete examples, show that:

(i) X is σ(X₁, X₂,€¦)-measurable.

(ii) X is not σ(X₁, X₂,€¦)-measurable.

(iii) If

Show that the X_ns and X can be modified into X_ns and X€™, so that

Pointwise, X€™ is σ(X€™₁, X€™₂,€¦.)-measurable, and

(As a consequence, instead of the X_ns and X one could use the X_ns and X€™, without loss of generality, and also ensure that X€™ is σ(X€™₁, X€™₂,€¦)-measurable.

(iv) Consider the measurable space (W, A, μ), and suppose that, for some w₀ ÃŽ W, {w₀} ÃŽ A and μ ({w₀}) = 0, Define X_n (w) = 0 on {w₀}. And X_2n€“1(w₀) = 2, X_2n (w₀) = 3, n ³ 1; and X(w) = on {w₀}^c, X(w₀) = 1.

Then verify that

Furthermore, modify the X_ns and X as indicated in part (iii), so that the conclusions of that part hold.

a.e. . + X. 0