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1d
accepted If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
1d
awarded  Caucus
1d
comment If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
Thank you, but I do not see what should be missing in my proof.
1d
comment If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
I am still confused, we have $$ || F ||_2^2 = \int_a^b |F(x)|^2 d x \le \int_a^b ||f||_2^2 (x - a) d x $$ by monotonicity of the integral, and so this approach seems to work for me. And yes, $F(x) \in \mathbb R$, so writting $||F(x)||_2^2$ is strictly the norm of a constant, but that is not what I meant there... (btw your purely Cauchy-Schwarz proof would be of interest)
2d
comment If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
After some time I found a simpler approach, just using Cauchy-Schwarz. Am I missing something? It goes like this: Let $a < x <b$, then \begin{align*} |F(x)| & = |\int_a^b f(y) 1_{[a,x]} dy| \\ & \le \int_a^b |f(y)| 1_{[a,x]}(y) dy \\ & = \langle |f|, 1_{[a,x]} \rangle \\ & \le ||f||_2 \cdot || 1_{[a,x]} || \\ & = ||f||_2 (x - a) \end{align*} and so $ ||F(x)||_2^2 \le \int_a^c ||f||_2 (x-a)^2 dx \in L_2(a,b)$, where the last term is contained in $L_2(a,b)$ because it is simply a polynomial. Seems way simpler then applying the general Minkowski inequality, but maybe I overlooked something?
Dec
16
accepted Definition of conditional probabiliy as function dependent on $\sigma$-Algebra
Dec
16
comment Definition of conditional probabiliy as function dependent on $\sigma$-Algebra
Thanks, I will look it up, and btw "the Radon-Nikodym theorem have just been proved..." lucky coincidence for him^^
Dec
16
comment Definition of conditional probabiliy as function dependent on $\sigma$-Algebra
Yes, and from this I derived a function $\Omega \to [0,1]$ by $P(A | \{ \omega \})$, so here I derived two functions, one from $\mathcal F$ and one from $\Omega$, maybe not that clearly stated but thats what I meant, because in the general definition it is also defined as a function $\Omega \to [0,1]$
Dec
15
asked Definition of conditional probabiliy as function dependent on $\sigma$-Algebra
Dec
14
comment If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
Thank you, now I solved it. But let me note that I used a more general version of Minkowski then the usual one $||f + g||_p \le ||f||_p + ||g||_p$ which I did not know beforehand. You can find it on wikipedia under the headline "Minkowski's integral inequality".
Dec
13
asked If $f \in L_2(a,b)$, then $\int_a^x f(y) dy \in L_2(a,b)$?
Dec
7
comment Solving differential equation by weak formulation and minimizing a functional
Because it is given that I have to look in this function space. The Euler-Lagrange equation is the original ODE I guess, or not?
Dec
6
awarded  Popular Question
Dec
5
asked Solving differential equation by weak formulation and minimizing a functional
Nov
26
revised What does instability mean and examples, boundary condition
edited body
Nov
24
revised What does instability mean and examples, boundary condition
added 1 character in body
Nov
24
asked What does instability mean and examples, boundary condition
Nov
13
accepted How are rational exponents defined in groups?
Nov
13
asked How are rational exponents defined in groups?
Nov
5
accepted Showing countable additivitiy of Lebesgue measure