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====== Lecture 5 ====== | ====== Lecture 5 ====== | ||
+ | $\gdef\mcal{\mathcal{M}}$ | ||
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Welcome back to in-person instruction. I will continue type in here as a way to prepare for class. | Welcome back to in-person instruction. I will continue type in here as a way to prepare for class. | ||
After a long toil of last two weeks, we have established the existence of measurable sets and Lebesgue measure on $\R^n$. We know open sets and closed sets are measurable, and countable operations won't take us away from measurable sets. The Lebesgue measure on measurable sets satisfies all the intuitive properties that you wish it has. | After a long toil of last two weeks, we have established the existence of measurable sets and Lebesgue measure on $\R^n$. We know open sets and closed sets are measurable, and countable operations won't take us away from measurable sets. The Lebesgue measure on measurable sets satisfies all the intuitive properties that you wish it has. | ||
- | Next, we will consider | + | After an actual reading of Tao's 7.5, I decided that it is a bit misleading (especially the part that composition of measurable functions are not automatic measurable (but will turns out to be so after some work). Let's first review what should be true in general. |
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+ | ===== Abstract Measure Space ===== | ||
+ | Let $S$ be a set, and $2^S$ be the set of subsets in $M$. | ||
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+ | ** $\sigma$-algebra **: Let $\mcal_S$ be a subset of $2^S$. | ||
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+ | We refer to the pair of a space and a $\sigma$-algebra, | ||
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+ | ** measure on $(S, \mcal_S)$ **: A measure is a function | ||
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+ | The triple $(S, \mcal_S, \omega)$ | ||
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+ | ** measurable function | ||
+ | a function | ||
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+ | This may reminds you of the definition of topological spaces and continuous | ||
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+ | Hence by definition, composition of measurable sets are measurable. Why we care about composition of measurable set? It is useful in ergodic theory, which is about iterations of $f^n$, where $f: X \to X$ is measurable. | ||
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+ | If $S$ happens to also be a topological space, with $\tau_S$ denote the set of open sets, we may define **Borel $\sigma$-algebra $\mathcal{B}_S$ **, which is the smallest $\sigma$-algebra of $2^S$ that contains $\tau_S$. If a subset of $S$ is in the Borel sigma-algebra, we call it a Borel set. In particular, | ||
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+ | Now, you may ask, consider $\R$ with the usual topology, are Borel sets equivalent with Lebesgue measurable set? Not quite. They may differ by a measure zero subset (called null set, or zero set). | ||
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+ | ===== Pugh 6.2: construction of measure ===== | ||
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+ | Let's turn to Pugh now. We first need to revisit Theorem 6.5, page 389. Let $S$ be any set. One can construct a $\sigma$-algebra and a measure on it, starting from any outer-measure $\omega: 2^S \to [0, \infty]$. An outer-measure $\omega$ on $S$ is any such function that satisfies $\omega(\emptyset)=0$, | ||
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+ | From outer-measure $\omega$, one can define measurable set on $S$, using Caratheory criterion. Namely, $E$ is measurable | ||
+ | We need to show that, measurable set forms a $\sigma$-algebra. The proof is no different than Tao 7.4.8. | ||
+ | In short, Pugh's theorem 5 is a 'free upgrade' | ||
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+ | One statement worth emphasizing is that, " | ||
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+ | Hyperplanes | ||
+ | $$ \{0\} \times \R = \cup_{n=1}^\infty (-\epsilon 2^{-2n-2}, \epsilon 2^{-2n-2}) \times (-2^n, 2^n) $$ | ||
+ | where the sum of area of boxes is less than $\epsilon$. | ||
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+ | ===== Pugh 6.4: Regularity ===== | ||
+ | Our goal here is to prove that, any Lebesgue measurable set is a Borel set plus or minus a null-set. More precisely. $E$ is Lebesgue measurable, if and only if there is a $G_\delta$-set (countable intersection of open) $G$, and an $F_\sigma$-set, | ||
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