If I understand you correctly, simply test if the maximum value in the list is Positive:
Positive @ Max @ a
Speed comparison with other methods that were posted:
timeAvg =
Function[func,
Do[If[# > 0.3, Return[#/5^i]] & @@ Timing@Do[func, {5^i}], {i, 0, 15}],
HoldFirst];
a = RandomInteger[{-1*^7, 2}, 1*^7];
MemberQ[a, _?Positive] // timeAvg
Total@UnitStep[-a] =!= Length@a // timeAvg
Positive@Max@a // timeAvg
0.593
0.0624
0.01148
Early-exit methods
Although very fast, especially with packed lists, the method above does scan the entire list with no possibility for an early exit when a positive elements occurs near the front of the list. In that case a test that does not scan the entire list may be faster, such as the one that R.M posted. Exploring such methods I propose this:
! VectorQ[a, NonPositive]
Unlike MemberQ, VectorQ does not unpack a packed list.
Timings compared to MemberQ and Max, first with an early positive appearance:
SeedRandom[1]
a = RandomReal[{-1*^7, 1000}, 1*^7];
Positive @ Max @ a // timeAvg
! VectorQ[a, NonPositive] // timeAvg
MemberQ[a, _?Positive] // timeAvg
0.008736 0.00013984 0.2528
(Most of the MemberQ time is spent unpacking the list.)
Then no positive appearance (full scan):
a = RandomInteger[{-1*^7, 0}, 1*^7];
Positive @ Max @ a // timeAvg
! VectorQ[a, NonPositive] // timeAvg
MemberQ[a, _?Positive] // timeAvg
0.01148 1.544 2.528
Finally a mid-range appearance of a positive value in an unpacked list:
a = RandomReal[{-50, 0}, 1*^7];
a[[5*^6]] = 1;
Positive @ Max @ a // timeAvg
! VectorQ[a, NonPositive] // timeAvg
MemberQ[a, _?Positive] // timeAvg
0.212 0.702 1.045