Intereting Posts

Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?
Zeros of analytic function and limit points at boundary
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?
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Solve the following equation for x and y:
In combinatorics, how can one verify that one has counted correctly?
Universal cover of $\mathbb R^2\setminus\{0\}$
Selection problem: how to solve?
Is $\|x\| = \| \overline{x} \|$ in an inner product space?
Prove that $\frac{n+1}{2} \leq 2\cdot\sqrt{2}\cdot\sqrt{2}\cdot\sqrt{2}\cdots\sqrt{2}$
How find the maximum value of $|bc|$
uniform random point in triangle
Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $
Proof – Inverse of linear function is linear

Let $E=\cup_{i=1}^\infty A_i$ where the measure of $A_i$ is zero. How can I conclude that $E$ has measure zero?

- Generating the Borel $\sigma$-algebra on $C()$
- What is wrong in this proof: That $\mathbb{R}$ has measure zero
- Show that: $\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$
- T-invariant sub-sigma algebra
- Stone's Theorem Integral: Basic Integral
- Notation question: Integrating against a measure
- Random Variable Inequality
- Constructively generating a sigma algebra
- Weird measurable set
- Equivalent Definition of Measurable set

If your measure $\mu$ is sub sigma additive you can surely say that

\[ \mu(E)\leq \sum_{i=1}^\infty \mu (A_i)=0\]

If it’s not the statement doesn’t need to be true.

Hint: Assuming the measure is Lebesgue measure, for each $A_i$, you can find an open set $O_i$ such that $A_i\subseteq O_i$ and the measure of $O_i$ is less than $\varepsilon/2^i$. If not, as long as the measure is subadditive, you can follow Dominic Michaelis’ approach.

It follows by sigma-aditivity.

$$m\bigg(\bigcup_{i\in\mathbb {N}}A_{i}\bigg)\leq\sum_{i\in\mathbb{N}}m(A_{i})=0$$

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- What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?