= Solution
Use a <standard Borel probability space> so that <conditional measures> can be realized by measures. The <disintegration theorem for a probability measure> states the following. For a measurable map $\pi:X\to Y$ between standard Borel spaces and $\nu=\pi_*\mu$, there is a <measurable probability kernel> $y\mapsto\mu_y$, unique for $\nu$-almost every $y$, such that
$$
\boxed{\mu(A)=\int_Y\mu_y(A)\,d\nu(y),\qquad
\mu_y(\pi^{-1}\{y\})=1\quad\text{for }\nu\text{-almost every }y.}
$$
Measurability means $y\mapsto\mu_y(A)$ is measurable for every Borel set $A$. More generally
$$
\int_Xf\,d\mu=\int_Y\left(\int_Xf\,d\mu_y\right)d\nu(y),
\qquad
\mathbb E_\mu[f\mid\pi^{-1}\mathcal B_Y](x)
=\int_Xf\,d\mu_{\pi(x)}
$$
for integrable $f$, with equality of the last expression <almost everywhere>. The $\mu_y$ are the <conditional measures of a factor>. This also defines <conditional measures> for a countably generated sub-sigma-algebra, or a measurable partition represented by such a map. Sigma-algebras are considered modulo null sets where necessary.
For an invariant <probability measure>, disintegrate over the <invariant sigma-algebra> $\mathcal I$. On a standard probability space it can be represented, modulo null sets, by a countably generated measurable factor $\pi$. One justification for this countability is the separability of $L^2$: a countable dense family of $\mathcal I$-measurable functions generates $\mathcal I$ modulo null sets. \b[The resulting <conditional measures> are the <ergodic components>]:
$$
\boxed{\mu=\int_Y\mu_y\,d\nu(y),\qquad
\mathbb E_\mu[f\mid\mathcal I](x)=\int f\,d\mu_{\pi(x)}.}
$$
Their invariance is one of the permitted facts. It remains to prove that almost every component is <ergodic>, rather than simply invariant.
Choose a <countable generating algebra> $\mathcal A$ generating the Borel sigma-algebra of $X$. Apply the <Birkhoff ergodic theorem> simultaneously to its indicators. For $\mu$-almost every $z$ and all $A\in\mathcal A$,
$$
\frac1N\sum_{n<N}\mathbf1_A(T^nz)
\longrightarrow\mathbb E_\mu[\mathbf1_A\mid\mathcal I](z)
=\mu_{\pi(z)}(A).
$$
Disintegration transfers this common full-measure set to $\mu_y$-almost every $z$, for $\nu$-almost every $y$. On that fiber the <conditional expectation> is constant and equals $\mu_y(A)$. Thus, simultaneously for every $A\in\mathcal A$,
$$
\frac1N\sum_{n<N}\mathbf1_A(T^nz)\longrightarrow\mu_y(A)
\quad\text{for }\mu_y\text{-almost every }z.
$$
Fix such a $y$, for which $\mu_y$ is also invariant. Applying the <Birkhoff ergodic theorem> to the system with measure $\mu_y$ identifies these limits with
$$
\mathbb E_{\mu_y}[\mathbf1_A\mid\mathcal I_y]=\mu_y(A).
$$
Finite linear combinations of the algebra's indicators are dense in $L^2(\mu_y)$, by the <Monotone class theorem>. Since <conditional expectation> is an $L^2$ contraction, its projection onto invariant functions consequently sends every $L^2(\mu_y)$ function to its constant integral. In particular, for any $\mu_y$-invariant set $D$,
$$
\mathbf1_D=\mathbb E_{\mu_y}[\mathbf1_D\mid\mathcal I_y]
=\mu_y(D)\quad\mu_y\text{-almost everywhere},
$$
and so $\mu_y(D)\in\{0,1\}$. This proves \b[almost every component is <ergodic>]. The <countable-test proof of ergodicity of conditional components> avoids taking an invalid intersection of uncountably many full-measure sets.
For rotation by $1/5$, take the quotient map $\pi(x)=5x\pmod1$. Its fibers are exactly the five-point orbits. The <ergodic components of a rational circle rotation> are therefore
$$
\boxed{\mu_y=\frac15\sum_{j=0}^4
\delta_{(y+j)/5},\quad y\in[0,1),\qquad
m=\int_0^1\mu_y\,dy.}
$$
Equivalently the component through $x$ is $\frac15\sum_{j=0}^4\delta_{x+j/5}$. To check the disintegration, for bounded measurable $f$ use the change of variables on the five consecutive fifth-intervals:
$$
\int_0^1\frac15\sum_{j=0}^4f\!\left(\frac{y+j}{5}\right)dy
=\sum_{j=0}^4\int_{j/5}^{(j+1)/5}f(x)\,dx
=\int_0^1f(x)\,dx.
$$
The rotation cyclically permutes the five atoms, preserving their equal masses. A set invariant modulo $\mu_y$ must contain all or none of this finite orbit, since every atom has positive mass. Thus each $\mu_y$ is <ergodic>. Finally, the <invariant sigma-algebra> is the pullback under $\pi$: a function invariant under translation by $1/5$ is constant on each orbit and is a measurable function of $5x\pmod1$. This confirms that these are the <conditional measures> over $\mathcal I$, not merely some decomposition into invariant measures.
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