You cannot "measure" your lactate threshold without scientific instruments, but you can get close to knowing what it is by the below info. ...and yes, you need an HR (heart rate) monitor. Here is a link to how and what of lactate threshold (LT) testing and figuring "yours" out.

http://www.lactate.com/pitesbas.html

The lactate threshold is different for the bike, swim and run, as is your HRmax for each; they get closer together as one gets fitter as well. But esentialy the LT is at about 80-85% of the HRmax for each event. I.e., if your HRmax for the run is (e.g., 200 beats per min.(bpm)) then the LT is around 160-170bpm. If the HRmax is 185bpm for the bike, then LT for the bike would be approx. 148-157bpm. Best thing is to figure out your HRmax for each event and then use the below information to figure out yours.

Read this:

http://www.coachgordo.com/gtips/endurance_essentials/aerobic_threshold_summary.html

Use the following to figure out your zones for the bike and run after you figure out your HRmax. for both the bike and run. Btw, the bike HRmax. is usually 20-25bpm lower than the run (rough est.):

HRzones by Joe Friel: Based on Lactate Threshold Heart Rate (LTHR)
LTHR = ~80-85% of HRmax.

Zone: Percentage of LTHR
1 65% - 81% Recovery
2 82% - 88% Aerobic
3 89% - 93% Tempo
4 94% - 100% SubThreshold
5A 101% - 102% SuperThreshold
5B 103% - 105% Aerobic Capacity
5C 106%+ Anaerobic Capacity

Additionally, do a search on this forum. There are many things to help you figure out the LT and how to train.