# lebesgue measure/Measurable sets

Question is :

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

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

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.

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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.