Week 3 Worksheet

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1. Fix a set $X$. A lambda system on $X$ is any subset of $\mathcal{P}(X)$ that contains $\varnothing$, is closed under complements, and is closed under countable disjoint unions. Fix measures $\mu$ and $\nu$ on a measurable space $(X,\mathcal{B})$. Prove that $$\{B\in \mathcal{B}:\mu (B)=\nu (B)\}$$ is a lambda system.
2. Use the π-𝜆 theorem to prove that if two Borel measures assign the same value to all intervals then they are equal.
3. How many Lebesgue measurable sets are there?
   1. Prove that if $E\in \mathsf{Leb}(\mathbb{R})$ has $\lambda (E)=0$ and $F\subset E$ then $F$ is in $\mathsf{Leb}(\mathbb{R})$.
   2. Conclude that every subset of the middle-thirds Cantor set $\mathcal{C}$ belongs to $\mathsf{Leb}(\mathbb{R})$.
4. Fix a measure space $(X,\mathcal{B},\mu )$. Suppose $n\mapsto f_n$ is a sequence of non-negative Borel measurable functions satisfying $$f_1(x)>f_2(x)>f_3(x)>\cdots$$ for all $x\in X$. Prove that $$\underset{n\to \mathrm{\infty }}{lim}\int f_n\mathsf{d}\mu =\int \underset{n\to \mathrm{\infty }}{lim}{f}_n\mathsf{d}\mu$$ whenever the integral of $f_1$ is finite. Why is this last assumption necessary?
5. Fix a measure space $(X,\mathcal{B},\mu )$ and $f:X\to [0,\mathrm{\infty }]$ measurable. Assume $$\int f\mathsf{d}\mu =0$$ and prove that $\{x\in X:f(x)>0\}$ has zero measure.
6. Evaluate $\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \int 1_{[0,n]}(x)\cdot {(1-{\displaystyle \frac{x}{n}})}^n\cdot e^{x/2}\mathsf{d}\lambda (x)}$
7. Fix a measure space $(X,\mathcal{B},\mu )$ and a measurable function $f:X\to [0,\mathrm{\infty }]$. Define $\nu$ on $(X,\mathcal{B})$ by $$\nu (E)=\int 1_{E}f\mathsf{d}\mu$$ for all $E\in \mathcal{B}$. Is $\nu$ a measure?
8. Let $\mu$ be counting measure on $\mathbb{N}$. Interpret the monotone convergence theorem in this special case. Looking through our proof of the monotone convergence theorem, is there any circular reasoning?
9. Fix a measure space $(X,\mathcal{B},\mu )$. Prove Fatou's lemma which states that $$\int \underset{n\to \mathrm{\infty }}{lim inf}{f}_n\mathsf{d}\mu \le \underset{n\to \mathrm{\infty }}{lim inf}\int f_n\mathsf{d}\mu$$ for any sequence of functions $f_n:X\to [0,\mathrm{\infty }]$.
10. For each of the following sequences of functions from $\mathbb{R}$ to $\mathbb{R}$ determine its pointwise limit, whether the limit of the integral equals the integral of the limit, whether the monoton convergence theorem applies, and whether Fatou's lemma applies.
    1. $f_n(x)={\displaystyle \frac{1}{n}}\cdot 1_{(0,n]}$
    2. $f_n(x)=n\cdot 1_{(1/n,2/n]}$
11. Show that the monotone convergence theorem is a consequence of Fatou's lemma.