# Tag Info

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### The set of the points where a measurable function is continuous is measurable

The continuity set of a real-valued function is $G_{\delta}$ i.e., a countable intersection of open sets. Open sets are measurable. Countable intersections of measurable sets are measurable.

### Find a sequence of $Lˆ1$-summable functions with $|f_n(x)| \leq 1$ and some other properties

A simple example, which you may have had in mind with your example, are rectangles that are getting wider and wider: $$f_n(x) := χ_{(0,n]}(x) \cdot \frac{1}{n}.$$ The area is obviously always 1 ...
• 212
1 vote
Accepted

### Find a sequence of $Lˆ1$-summable functions with $|f_n(x)| \leq 1$ and some other properties

When you are faced with these problems, always look for easy solutions first. The easiest example of integrable functions are $\frac{1}{x^n}$, however there are two issues: They are not integrable ...
• 3,392

• 174k
1 vote

### $L^\infty(\Omega)$ is dense in $L^{p,\infty}(\Omega)$ if $\Omega$ is compact

This density is not true. Let $\Omega=(0,+1)$. Let $p\in (0,\infty)$ and define $$f(x) = x^{-\frac1p}.$$ Then $$\mu(\{x:\ |f(x)| > t\} ) = t^{-p}$$ for $t\ge1$, so that \$\|f\|_{L^{p,\infty}} = ...
• 50.3k

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