# Real Analysis Problems

## Problem 1

Suppose $\mu$ is an outer measure on $X$. Prove that the following are equivalent:

1. The measure $\mu$ is $\sigma$-finite.
2. There exists an increasing sequence $X_1 \subset X_2 \subset \cdots$ of $\mu$-measurable sets such that $X = \cup_{k=1}^\infty X_k < \infty$ for each $k \in \mathbb{N}$.
3. There exists a disjoint sequence $Z_1,Z_2,\cdots$ of $\mu$-measurable sets such that $X = \cup_{k=1}^\infty Z_k$ and $\mu(Z_k) < \infty$ for each $k \in \mathbb{N}$.

## LaTeX

Upmath converts LaTeX equations in double-dollars $:$ax^2+bx+c=0$. All equations are rendered as block equations. If you need inline ones, you can add the prefix \inline:$\text{\inline } p={1\over q}$. Place big equations on separate lines:$x_{1,2} = {-b\pm\sqrt{b^2 - 4ac} \over 2a}.$In this case the LaTeX syntax will be highlighted in the source code. You can even add equation numbers (unfortunately there is no automatic numbering and refs support): $|\vec{A}|=\sqrt{A_x^2 + A_y^2 + A_z^2}.$ It is possible to write Cyrillic symbols in \text command:$Q_\text{плавления}>0\$.

One can use matrices:

$T^{\mu\nu}=\begin{pmatrix} \varepsilon&0&0&0\\ 0&\varepsilon/3&0&0\\ 0&0&\varepsilon/3&0\\ 0&0&0&\varepsilon/3 \end{pmatrix},$

integrals:

$P_\omega={n_\omega\over 2}\hbar\omega\,{1+R\over 1-v^2}\int\limits_{-1}^{1}dx\,(x-v)|x-v|,$

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END OF TEST

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