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--- math105-s22:notes:lecture_15 [2022/03/08 00:27]
pzhou
+++ math105-s22:notes:lecture_15 [2022/03/08 00:30]
pzhou
@@ Line 27: / Line 27: @@
 If $f$ is not bounded, then we can split $f$ to be a bounded part and an unbounded part. For any $n>0$, we can define $f = f_n + g_n$, where $f_n = \max (f, n)$. Then, as $n \to \infty$,  $g_n \to 0$ a.e., and by DCT, $\int g_n \to 0$. Pick $N$ large enough, such that $\int g_N < \epsilon /2$. Then, let $\delta = (\epsilon/2) / N$. Then, for any countable disjoint open intervals $\{(a_i, b_i)\}$ with length $< \delta$, we have
-$$ \sum_i \int_{a_i}^b_i f(t) dt \leq \sum_i \int_{a_i}^b_i f_N(t) dt +  \sum_i \int_{a_i}^b_i g_N(t) dt \leq N \delta + \int_a^b g_N \leq \epsilon/2 + \epsilon/2 = \epsilon $$
+$$ \sum_i \int_{a_i}^{b_i} f(t) dt \leq \sum_i \int_{a_i}^{b_i} f_N(t) dt +  \sum_i \int_{a_i}^{b_i} g_N(t) dt \leq N \delta + \int_a^b g_N \leq \epsilon/2 + \epsilon/2 = \epsilon $$
 Done for (a).
@@ Line 38: / Line 38: @@
 Since $\epsilon$ is arbitrary, we do get $H(a)=H(b)$.
+-------
+I will leave Pugh section 10 for presentation project.

Lecture Notes