Invariant measure ($T_{\ast }\mu =\mu$)

For a measurable transformation $T$, a measure is invariant if the pushforward measure equals the original measure, or equivalently $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. This is the measure used in a measure-preserving system.

A multifractal measure has different local scaling behavior at different points. For a measure $\mu$ on a metric space, a basic local exponent, when the limit exists, isIt describes how rapidly mass near $x$ shrinks with radius. The Hausdorff dimension of each level set $\{x:\alpha (x)=a\}$ gives a spectrum rather than a single dimension. The word does not require the measure to be invariant, although many examples come from invariant measures of dynamical systems.

For a concrete construction, repeatedly split $[0,1]$ into halves, giving relative mass $p$ to the left half and $1-p$ to the right, with $0<p<1$. A level-$n$ interval with $k$ left choices has length $2^{-n}$ and mass $p^k(1-p{)}^{n-k}$. Along a nested sequence with $k/n\to q$, its cylinder scaling exponent is $-[q\log p+(1-q)\log (1-p)]/\log 2$. When $p\ne 1/2$, different digit frequencies produce different exponents. This is the mechanism by which uneven concentration of mass leads to several scaling laws.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.