实分析期末试题

终于考完了期末考试，漫漫长路，加油吧！

👇为试题：

1. 叙述有界变差函数与绝对连续函数的定义，并说明两者的关系.

2. 设\(\nu,\mu\)是两个Lebesgue-Stieltjes测度，若\(\nu \ll \mu\), 证明：若集合\(E\)是\(\mu\)可测，则\(E\)也是\(\nu\)可测.

3. 证明：若\(m(E)<+\infty\)(这里\(m\)表示Lebesgue测度)，则\(f_n\)在\(E\)上几乎处处收敛与依测度收敛等价.

4. （本题为课本定理）Let \(\mathcal{A} \subset \mathcal{P}(X)\) be an algebra, \(\mu_0\) a premeasure on \(\mathcal{A}\), and \(\mathcal{M}\) the \(\sigma\)-algebra generated by \(\mathcal{A}\). There exists a measure \(\mu\) on \(\mathcal{M}\) whose restriction to \(\mathcal{A}\) is \(\mu_0\). If \(\mu_0\) is finite , then \(\mu\) is the unique extension of \(\mu_0\) to a measure on \(\mathcal{M}\).

5. (1) 若\(f\in L^1\),证明集合\(\{x:f(x)\ne 0\}\)是\(\sigma\)-有限的.

   (2) 设\(E\)是\(\mathbb{R}\)上的Lebesgue可测集，若对任一可测集\(E\)，都有

   \[\int_{E} f(x) \,\mathrm{d}x=0, \]

   证明：\(f(x)=0\) a. e.

6. (1) 若\(f\)是一个Lebesgue可测函数，则存在函数\(g\)为Borel可测函数，使得\(f\)与\(g\)几乎处处相等.

   (2) 证明：若\(f(x)\)是\(\mathbb{R}^n\)上的可测函数，证明\(f(x-y)\)是\(\mathbb{R}^{n} \times \mathbb{R}^{n}\)上的可测函数.

7. 假设 \(\nu(E) = \int f d\mu\)，其中 \(\mu\) 是一个正测度，\(f\) 是一个 \(\mu\) 可积函数. 请根据 \(f\) 和 \(\mu\) 描述 \(\nu\) 的 Hahn 分解，以及 \(\nu\) 的正变、负变和全变差.