Convergence Theorems

Prove the following generalization of the Dominated Convergence Theorem. Let \((X,\mathcal {M},\mu )\) be a measure space. Suppose \(f_n,g_n,f,g:X\to [0,\infty )\) are measurable and have finite integrals, \(f_n\to f\), \(g_n\to g\) (pointwise convergence), \(f_n\le g_n\), \(\int g_n\to \int g<\infty \). Then \(\int f_n\to \int f\).

State the Lebesgue dominated convergence theorem. Show that the converse is not true. That is, there is a sequence of functions that satisfy the conclusion of the Lebesgue dominated convergence theorem but not its assumption.

1. State the Monotone Convergence Theorem.
2. Let \(f : [0, \infty ) \to \mathbb {R}\) defined by \(f(x) = e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral. You may use the fact that if a function \(f : [0, \infty ) \to \mathbb {R}\) is 0 outside an interval \([a, b]\) and \(f|_{[a,b]} : [a, b] \to \mathbb {R}\) is Riemann integrable then \(f\) is Lebesgue integrable and the Lebesgue integral of \(f\) equals the Riemann integral of \(f|_{[a,b]}\).

Find the Taylor series of the function \[ f(z)=\frac {z}{2+z} \] at the point \(a=1\). What is the radius of convergence of that series?

Suppose that \(f\) is holomorphic in the open disc of radius \(r\), centered at \(z_0\), but does not have an analytic extension to any neighborhood of the closure of this disc. Show that the radius of convergence of the power series expansion for \(f\) centered at \(z_0\) is \(r\).