6210-2024-08

Let

Show that \(A\) has Lebesgue measure 0 but

Suppose \(f_i:\mathbb {R}\to \mathbb {R}\) is measurable for \(i=1,2,\cdots \). Is the set \[ \{x\in \mathbb {R}\mid 2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \]

measurable? A number \(t\) is a limit point of a sequence if some subsequence converges to it.

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 \(\int f_n\to \int f\).

Let \(\mu \) and \(\nu \) be two finite positive measures on a measurable space \((X,\mathcal {M})\). Suppose for every \(\epsilon >0\), there exists \(E\in \mathcal {M}\) such that \(\mu (E)<\epsilon \) and \(\nu (X\setminus E)<\epsilon \). Show that \(\mu \perp \nu \) (\(\mu \) and \(\nu \) are mutually singular).

Let \((X,\mathcal {M},\mu )\) be a measure space with \(\mu (X)<\infty \). Show that if \(1\le q< p\le \infty \) then \(L^q(X)\supseteq L^p(X)\). Moreover, if \(X\) contains an infinite family of pairwise disjoint subsets of positive measure, then \(L^q(X)\supsetneq L^p(X)\).

Prove that the dual space of \((c_0,\|\cdot \|_{\sup })\) is \(\ell ^1\). Recall \[ c_0=\{v\in \mathbb {R}^{\mathbb {N}}:\lim _{j\to \infty }v_j=0\} \]

and \(\ell ^1=\{w\in \mathbb {R}^{\mathbb {N}}:\sum _j |w_j|<\infty \}\) with \(\|w\|=\sum _j |w_j|\).