Intereting Posts

How do people who study intensely abstract mathematics “imagine” or understand the concepts they are studying or being taught?
show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$
Representing IF … THEN … ELSE … in math notation
$f$ is irreducible $\iff$ $G$ act transitively on the roots
Number of $\sigma$ -Algebra on the finite set
How can one compute this simple infinite product?
Using Recursion to Solve Coupon Collector
Can an odd perfect number be divisible by $825$?
Is $z^{-1}(e^z-1)$ surjective?
Isomorphism of Vector spaces over $\mathbb{Q}$
Regularity of a domain – definition
IMO 2016 P3, number theory with the area of a polygon
Weird large K symbol
Why is empty set an open set?

It is rather straightforward to show that the sum of two measurable functions is also measurable. Therefore we can extend the logic to say that $\sum\limits_{i=1}^n f_i$ is measurable providing $f_i$ is measurable. However can we take $n\rightarrow\infty$ and say that $\sum\limits_{i=1}^\infty f_i$ is a measurable function?

- The finite-dimensional distributions of a centered Gaussian process are uniquely determined by the covariance function
- $\lim_{n\rightarrow \infty } \int_{a}^{b}g_{n}(x)\sin (2n\pi x)dx=0$ where $g_{n}$ is uniformly Lipschitz
- Are simple functions dense in $L^\infty$?
- Can sets of cardinality $\aleph_1$ have nonzero measure?
- Do these $\sigma$-algebras on second countable spaces coincide?
- Various kinds of derivatives
- Conditional Expectation of Functions of Random Variables satisfying certain Properties
- Everything in the Power Set is measurable?
- Ultrafilters and measurability
- Counterexample for a non-measurable function?

As Daniel pointed out, the pointwise limit of measurable functions is measurable.

If we define $g_n=\sum_{i=1}^n f_i$, then as long as $\sum_{i=1}^{\infty} f_i = \lim_{n\to\infty}g_n$ converges, it defines a measurable function.

You need some assumptions here, otherwise your infinite sum will not converge, think of $f_n(x)=1$. See https://en.wikipedia.org/wiki/Dominated_convergence_theorem for details.

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