We introduce a new measure notion on small complexity classes (called $F$-measure),

based on martingale families,

that gets rid of some drawbacks of

previous measure notions:

it can be used to define dimension because martingale families can make money on all strings,

and it yields random sequences with an equal frequency of $0$'s and $1$'s.

As applications to $F$-measure,

we show that for almost every language $A$ decidable in subexponential time, $\p^A =\bpp^A $.

We show that almost all languages in \pspace\ do not have small non-uniform complexity.

We compare $F$-measure to previous notions and prove that martingale families

are strictly stronger than $\Gamma$-measure, we also discuss

the limitations of martingale families concerning finite unions.

We observe that all classes closed under polynomial many-one reductions have measure zero in \expc\ iff they have

measure zero in $\subexp$.

We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension

on \p ,

which meets the intuition behind Lutz's notion. We show that $\p$-dimension lies between finite-state dimension

and dimension on $\e$.

We prove an analogue to the Theorem of Eggleston in \p ,

i.e. the class of languages whose characteristic sequence contains $1$'s with frequency $\alpha$,

has dimension the Shannon entropy of $\alpha$ in \p .

We introduce a new measure notion on small complexity classes (called F-measure), based on martingale families,

that get rid of some drawbacks of previous measure notions:

martingale families can make money on all strings,

and yield random sequences with an equal frequency of 0's and 1's.

As applications to F-measure,

we answer a question raised in a paper by Allender and Strauss, improving their result to:

for almost every language A decidable in subexponential time,

P^A =BPP^A.

We show that almost all languages in PSPACE require large circuits.

We compare F-measure to previous notions and prove that martingale families are strictly stronger than a previous measure notion on P known

as $\Gamma$-measure. We also discuss

the limitations of martingale families concerning finite unions.

We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have

measure zero in SUBEXP.

We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension on P,

which meets the intuition behind Lutz's notion.

We show that the class of PSPACE-languages

with small circuit complexity has dimension 0 in PSPACE;

and prove an analogue to

the Theorem of Eggleston in P,

i.e. the class of languages whose characteristic sequence contains 1's with frequency $\alpha$,

has dimension the Shannon entropy of $\alpha$ in P.