Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 3

Question

Math 6010 • Linear Models 1996 • Question 3

Let \((\Omega ,\mathcal F,P)\) be a probability space, let \(\mathcal G\) be a sub-\(\sigma \)-field of \(\mathcal F\), and let \(M\) be the closed subspace of \(L^2(\Omega ,\mathcal F,P)\) consisting of the \(\mathcal G\)-measurable functions in \(L^2\). Show that \(T(X)=\mathbb {E}[X\mid \mathcal G]\) coincides with the orthogonal projection of \(L^2(\Omega ,\mathcal F,P)\) onto \(M\).

Raw LaTeX

Let \((\Omega,\mathcal F,P)\) be a probability space, let \(\mathcal G\)
be a sub-\(\sigma\)-field of \(\mathcal F\), and let \(M\) be the closed subspace of
\(L^2(\Omega,\mathcal F,P)\) consisting of the \(\mathcal G\)-measurable functions in
\(L^2\). Show that \(T(X)=\E[X\mid \mathcal G]\) coincides with the orthogonal
projection of \(L^2(\Omega,\mathcal F,P)\) onto \(M\).

Topic Tags

L2 Convergence and Covariance Bounds Measure-Theoretic Probability Orthogonal Projections Uniform Integrability and Truncation Methods