  In this article, we combine the perspectives of density, entropy, and
multifractal analysis to investigate the structure of ergodic measures. We
prove that for each transitive topologically Anosov system $(X,f)$, each
continuous function $\varphi$ on $X$ and each $(a,h)\in \mathrm{Int}\{(\int
\varphi d\mu, h_\mu(f)):\mu\in M_f(X)\},$ the set $\{\mu\in M_f^e(X): (\int
\varphi d\mu, h_\mu(f))=(a,h)\}$ is non-empty and contains a dense $G_\delta$
subset of $\{\mu\in M_f(X): (\int \varphi d\mu, h_\mu(f))=(a,h)\}.$ Meanwhile,
combining the development of non-hyperbolic systems and cocycles we give a
general framework and use it to obtain intermediate entropy property of ergodic
measures with same Lyapunov exponent for non-hyperbolic step skew-products,
elliptic $\operatorname{SL}(2, \mathbb{R})$ cocycles and robustly
non-hyperbolic transitive diffeomorphisms. Moreover, we get generalized results
on multiple functions and use them to obtain the intermediate Hausdorff
dimension of ergodic measures for transitive average conformal or
quasi-conformal Anosov diffeomorphisms, that is $\left\{\operatorname{dim}_H
\mu: \mu\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H \mu: \mu\in
M_f(M)\right\}.$ In this process, we introduce and establish a
'multi-horseshoe' entropy-dense property and use it to get the goal combined
with the well-known conditional variational principles. As applications, we
also obtain many new observations on various other quantitative spectrums
including Lyapunov exponents, first return rate, geometric pressure, unstable
Hausdorff dimension, etc.
