Weil's book Basic Number Theory, (1973, second edition) pp. 113 mentioned the Tamagawa measure on $k_\mathbb{A}$, where $k$ is a global field and $k_\mathbb{A}$ its ring of adeles (an old fashioned notation). So it's defined according to the following process:
First, we know that by definition the (additive) group of adeles is locally compact, Hausdorff. Then by the general theory of Haar measure, we can pick a unique $\alpha$ such that $\alpha(k_\mathbb{A}/k)=1$, which is defined to be the Tamagawa measure on the group of adeles. And by a computation carried out on pp. 112, we know that if we pick the self dual measures (with respect to Fourier transform) $\alpha_v$ on each place $v$ of $k$, then $\alpha=\prod \alpha_v$.
This is a nice result, and I know how to deal with it down to earth: you pick a "fundamental" open set on $k_\mathbb{A}$ of the form $U=\prod U_v$, where $U_v$ is an open set in $k_v$ and almost all $U_v=R_v$ (the ring of integers of $k_v$). Then the formula simply means $\alpha(U)=\prod \alpha_v(U_v)$ (the product is well-defined since all but finitely many are $1$).
OK. Now what makes me puzzled is the following: to be explicit and clearer, for example, $k=\mathbb{Q}$. You have prepared all $\alpha_p$ and $\alpha_\infty$ which are self-dual. Now I "adjust" $\alpha_2'=\lambda\cdot\alpha_2$ and $\alpha_3'=1/\lambda\cdot\alpha_3$ for some $\lambda>0$, and keep other $\alpha_v’=\alpha_v$. Then by definition we must have $\alpha'=\prod \alpha_v'$ is also the Tamagawa measure, since its value on a standard open set $U=\prod\mathbb{Z}_p\times [0,1]$ is the same as the original one.
This looks somewhat weird for me (as a beginner). So you can "adjust" (arbitrarily many!) finitely many local measures without change the Tamagawa measure itself. Is this true? (I'm not quite sure as a new comer.) If this is the case, then how to understand this phenomenon intuitively? Of course usually we shall construct the Tamagawa measure following the above standard process. But in general by this observation, we can't use Tamagawa measure to point out any local measure on finitely many places? Thanks a lot in advance for your explanation!
One salient point is that, since you can only really change all the local measures simultaneously by a global number $\lambda$, the product formula $\prod_v |\lambda|_v=1$ says that the global measure does not change.