# Fourier Transform of Even/Odd Complex Functions

For any complex-valued $$f in L^1(mathbb{R})$$, let’s define its Fourier transform $$hat{f}$$ with the following convention
$$hat{f}(omega) := int_{mathbb{R}} f(x) e^{-i omega x} dx$$

I would like a confirmation of the following:

• $$f$$ even $$Rightarrow$$ $$hat{f}$$ even
• $$f$$ odd $$Rightarrow$$ $$hat{f}$$ odd

To prove for example the first statement, I would argue that
$$hat{f}(omega) = int_{mathbb{R}} f(x) e^{-i omega x} dx = int_{mathbb{R}} f(-x) e^{i omega x} dx = int_{mathbb{R}} f(x) e^{i omega x} dx = int_{mathbb{R}} f(x) e^{-i (-omega) x} dx = hat{f}(-omega)$$
where the second equality is a consequence of the fact that the Lebesgue measure ($$lambda$$) satisfies $$lambda(A) = lambda(-A)$$, for every Borel set $$A$$ and the third is just using the hypothesis, i.e. that $$f(x) = f(-x)$$.

The second statement can be proved in the very same way.

Is the above correct?!?

I started having second thoguhts after seeing this heavily downvoted answer
Fourier transform of even/odd function
and especially its first comment.