I am trying to understand this proof of Lemma 4.8 of chapter 7 in page 365 of Real Analysis by Stein and Shakarchi.
Here, $f$ is assumed to be continuous with compact support in $R^d$, and $\hat{f}$ is its Fourier transform.
My question is, how is the change of variables justified? I feel like you can choose a $f$ such that $\hat{f}$ is not so “even”, by shifting $\hat{f}$ as necessary, so those two integrals over the positive reals and the negative reals are not the same. Link to an image of the proof
Edit: The line is $$ \int_{S^{d-1}} \int_0^\infty |\hat{f}(\lambda \gamma)|^2 \lambda^{d-1} d \lambda d \sigma(\gamma) = \int_{S^{d-1}} \int_{-\infty}^0 |\hat{f}(\lambda \gamma)|^2 |\lambda|^{d-1} d \lambda d \sigma(\gamma) $$
While typing this out, it occurred to me that is it because the integral over $S^{d-1}$ takes all directions, so changing $\lambda$ to $-\lambda$ does not matter?