Intereting Posts

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third-order nonlinear differential equation
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$\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but not free over $\mathbb{Z}$.
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Number of roots of polynomials in $\mathbb Z/p \mathbb Z $
How prove this sequence $S_{n}=,n\in N$ contains infinitely many composite numbers
Assume that $ f ∈ L()$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.
Example of a closed subspace of a Banach space which is not complemented?
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Prove that $a_n=1+\frac{1}{1!} + \frac{1}{2!} +…+ \frac{1}{n!}$ converges using the Cauchy criterion
Compact metrizable space has a countable basis (Munkres Topology)
Finding a diagonal in a trapezoid given the other diagonal and three sides

Question is :

Let $f,g$ be measurable real valued functions on $\mathbb{R}$ such that :

$$\int_{-\infty}^{\infty} (f(x)^2+g(x)^2)dx=2\int_{-\infty}^{\infty} f(x)g(x)dx$$

- If $f$ is Lebesgue integrable on $$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.
- Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{}$ almost everywhere.
- How can using a different definition for the integral be useful?
- Lebesgue integral basics
- How to find a measurable but not integrable function or a positive integrable function?
- Derivative of Fourier transform: $F'=F$

Let $E=\{x\in \mathbb{R} : f(x)\neq g(x)\}$ . Which of the followng statements are necessarily true?

- $E$ is empty set
- $E$ is measurable
- $E$ has lebesgue measure $0$
- For almost all $x\in \mathbb{R}$ we have $f(x)=0$ and $g(x)=0$

What all i could see is that second bullet and third bullet are probably correct. Because :

$$\int_{-\infty}^{\infty} (f(x)^2+g(x)^2)dx=2\int_{-\infty}^{\infty} f(x)g(x)dx$$

i.e.,

$$\int_{-\infty}^{\infty} (f(x)^2+g(x)^2)dx-2\int_{-\infty}^{\infty} f(x)g(x)dx=0$$

i.e.,

$$\int_{-\infty}^{\infty}(f(x)-g(x))^2dx=0$$

Though i have negative limits my function $(f(x)-g(x))^2$ is positive

So, I would see that $E=\{x\in \mathbb{R} : f(x)\neq g(x)\}$ is measurable and has measure $0$

Please tell me if what i have done is sufficient/clear.

- Darboux Theorem
- Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$.
- upper limit of $\cos (x^2)-\cos (x+1)^2$ is $2$
- Various kinds of derivatives
- Real Analysis, Folland Proposition 2.1
- What are the main uses of Convex Functions?
- Show $f$ is not $1-1$
- Show that the only tempered distributions which are harmonic are the the harmonic polynomials
- Is there a function with a removable discontinuity at every point?
- Proving if $\lim_{n\rightarrow\infty}a_n=L $ then $\lim_{n\rightarrow\infty} \frac{a_1+a_2+\cdots+a_n}n=L $

Yes, I think what you have done is right, provided that f and g are square integrable. And the last statement that f and g are zero is trivially false.

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