Since c^2 = 1, the elements e^+ = \tfrac{1+c}{2} and e^- = \tfrac{1-c}{2} are orthogonal idempotents in R[\ang{c}] summing to 1. They project M onto M^+ = e^+ M and M^- = e^- M, on which c acts as +1 and -1 respectively, giving the direct sum decomposition.

Work at finite level. Writing \GG_n = \Gal(F_n/\Q) and \GG_n^+ = \Gal(F_n^+/\Q), the quotient map of Galois groups induces a surjection \Zp[\GG_n] \twoheadrightarrow \Zp[\GG_n^+] which kills \Zp[\GG_n]^-, hence factors through a map \Zp[\GG_n]^+ \to \Zp[\GG_n^+]. Both sides are free \Zp-modules of rank (p-1)p^{n-1}/2, and the map sends a basis to a basis, so it is an isomorphism. Passing to the inverse limit over n gives \Lam(\GG)^+ \cong \Lam(\GG^+). (The formalisation proves bijectivity by a functional even-part section — \nu \mapsto \nu\circ(f \mapsto \tfrac12(f + f\circ c)) inverts the pushforward on the plus part — rather than the finite-level rank count, whose inverse-limit presentation of \Lam is deferred infrastructure; same map, different proof of bijectivity.)

By Lemma 10.1.1 write \mu = \mu^+ + \mu^- with \mu^\pm = \tfrac{1\pm c}{2}\mu; membership in \Lam(\GG^+) = \Lam(\GG)^+ means exactly \mu^- = 0. Since \chi(c) = -1, the change of variables x \mapsto cx shows \int_\GG\chi(x)^k\,d\mu^+ = \tfrac12\Big(\int_\GG\chi^k\,d\mu + (-1)^k\int_\GG\chi^k\,d\mu\Big), which vanishes for odd k and equals \int_\GG\chi^k\,d\mu for even k. The same computation with \mu^- shows \int_\GG\chi^k\,d\mu^- = 0 for all even k, while \int_\GG\chi^k\,d\mu^- = \int_\GG\chi^k\,d\mu for all odd k. Thus the hypothesis "\int_\GG\chi^k\,d\mu = 0 for all odd k\ge1" is equivalent to "\int_\GG\chi^k\,d\mu^- = 0 for all k\ge1". A measure on \Zpx$ whose moments \int x^k\,d\mu^- vanish for every k > 0 is itself zero, since these moments are the higher coefficients of its Mahler transform and the transform of a measure supported on \Zpx$ has vanishing constant term. Hence \mu^- = 0 precisely when the odd moments of \mu vanish.

By the interpolation property the moment \int_\GG\chi(x)^k\,d\zeta_p equals, up to the Euler factor at p, the value (1-p^{k-1})\,\zeta(1-k) of the Riemann zeta function at the trivial character. For odd k \ge 1 this vanishes: when k \ge 3 is odd, 1-k is a negative even integer, so \zeta(1-k) = 0 is a trivial zero of \zeta; and at k = 1 the p-adic Euler factor (1-p^{k-1}) = 1-p^0 = 0 itself kills the moment (here \zeta(0) = -\tfrac12 \neq 0, so it is the Euler factor that is responsible — the source's proof line "\zeta(1-k) = 0 for odd k \ge 1" overlooks this k = 1 case; erratum #13 of the formalisation's errata file). Hence all odd moments of \zeta_p vanish. Applying the plus-part criterion Lemma 10.1.3 to the measure ([g]-[1])\zeta_p \in \Lam(\GG), whose odd moments are (\chi(g)^k - 1)\int_\GG\chi^k\,d\zeta_p = 0, shows it lies in \Lam(\GG^+) for every g; therefore \zeta_p descends to a pseudo-measure on \GG^+.