This might be a question for mathoverflow instead of math.stackexchange, but it feels like there should be a simple answer to this and I haven't been able to work it out.
Suppose $M$ is a compact Riemannian manifold and $f : M \to M$ is a $C^1$ Anosov diffeomorphism (or more generally a nonuniformly hyperbolic diffeomorphism). Then for $\epsilon > 0$, at each point $x \in M$, there is a local foliation by local unstable manifolds: $$ W^u_\epsilon(x) = \left\{ y \in M : d\left(f^{-n}x, f^{-n}y\right) < \epsilon\right\}. $$ I'm working with the following definition (paraphrased from a few sources):
Suppose $m$ is the Riemannian volume (or Lebesgue measure) on $M$. The probability measure $\mu$ is an SRB measure if the map $f$ has nonzero Lyapunov exponents $\mu$-a.e., and the conditional measures $\mu_{u,x}$ on the leaves $W^u_\epsilon(x)$ of the local foliation are absolutely continuous with respect to $m_{u,x}$, the induced Riemannian leaf volume.
My question: What does it mean for $\mu_{u,x}$ to be a conditional measure? For a general probability space $(X, \mathcal A, \mu)$, if $A \in \mathcal A$ has $\mu(A) > 0$, then the conditional measure on $A$ is $\mu_A(B) = \mu(A \cap B)/\mu(A)$. But this definition makes no sense when $A$ is the leaf of the foliation, because since $\mu(M) = 1$, we must have $\mu(W^u_\epsilon(x)) = 0$. Is there a rigorous way to define $\mu_{u,x}$?
My attempt at an answer: At most points of $M$ (in fact at all points if $f$ is Anosov), there is also a foliation by local stable manifolds $$ W^s_\epsilon(x) = \left\{ y \in M : d\left(f^{n}x, f^{n}y\right) < \epsilon\right\}, $$ which is transverse to the local unstable foliation. Suppose $A \subset W^u_\epsilon(x) \cap B(x,\delta)$ is an open set in the subspace topology, where $B(x,\delta) := \{ y \in M : d(x,y) < \delta\}$. By slightly moving $A$ along the leaves of the local stable foliation, we create an open set $A' \subset B(x,\delta)$ of positive Lebesgue measure, and presumably, of positive $\mu$ measure. So, can we take the measure of the unstable "slice" of $A'$ that goes through $x$? Perhaps the argument involves local product structure for small $\epsilon, \delta$?
The terminology is quite confusing. What is meant by "conditional measure" in this context is often referred to as "disintegration measure," which is a related but distinct concept.
Let me make some general comments on disintegration measures-- what they are in the general setting and why they're useful-- and then afterwards I'll return to your question regarding SRB measures.
Disintegration measures
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and let $T : \Omega \to \mathbb R$ be a random variable. One often writes $\mathbb P( \cdot | \sigma(T))$ for the probability $\mathbb P$ conditioned on $\sigma(T)$, but it is natural to ask the following:
That is, can we realize the conditional probability $\mathbb P( \cdot | \sigma(T))$ as a family of probability measures $\mu_t$ supported on $\{ T = t\}$?
When $T$ takes finitely many or countably many distinct values a.s., the answer is trivial: of course! Here $\sigma(T)$ is generated by the countable partition $\{ \{ T = t_i\}, i =1,2,\cdots\}$ and the computations are straightforward. But when $T$ takes uncountably many values, the partition of $T$ into "level sets" $\{ T = t\}$ is necessarily uncountable and things get hairy.
This notion of disintegration measure was invented to get around this problem. The main theorem is that if $\Omega$ is assumed to have some topological structure and $\mathcal F$ is the Borel $\sigma$-algebra, then disintegration measures as above exist for any measurable $T : \Omega \to \mathbb R$.
Hopefully you see the connection between conditional measures and disintegration measures, explaining the confusing terminology.
One more comment: I don't really need a random variable to begin with to get disintegration measures. In the aforementioned setting, given any measurable partition $\zeta$ of my original probability space $\Omega$, I can find a family $(\mu_\alpha)_{\alpha \in \zeta}$ of disintegration measures with the property that
$$ \mathbb P(K | \sigma(\zeta)) = \mu_\alpha(K) \quad \text{ on the event } \alpha $$ for any Borel $K$. I'm not going to define here what a measurable partition is, except to say that $\zeta$ is measurable if it can be represented in the form $\zeta = \vee_{i = 1}^\infty \zeta_i$ for a collection $\zeta_i$ of countable partitions, where $\vee$ denotes the join (i.e., $\eta \vee \zeta = \{ \alpha \cap \beta : \alpha \in \eta, \beta \in \zeta\}$).
For more disintegration measures, I would recommend looking at "The Fundamentals of Measure Theory" by Rokhlin, as well as "Measure Theory through Dynamical Eyes" by Climenhaga and Katok and another paper on disintegration measures (precise name escapes me) by Chang and Pollard.
Defining the SRB property
In your context, to define the SRB property one often works with measurable partitions $\zeta$ subordinate to the unstable foliation, i.e., satisfying the following properties:
Then, one can define the family of disintegration measures $\mu_\alpha, \alpha \in \zeta$, each of which is a measure on some $u$-leaf. One checks now that if $\mu$ is SRB as above w.r.t. some partition $\zeta$, then it is SRB w.r.t. any partition $\zeta'$ subordinate to the unstable foliation.