First of all - what is the part number on the transformer - I can't quite read it from the photograph as the wires are in the way.
Most of the wires from the transformer go to the main logic board. They should enter the main logic board on a connector marked J8. Is this correct? My guess would be the two whites, the two blues and the black wire go to J8 (in addition to some red and black wires from the large capacitor).
You will also see two 'skinny brown' wires going from the transformer to the monitor.
Pins 1, 3 and 5 of J8 contain one set of voltages and pins 8 and 9 of J8 contain the other set of voltages.
Follow the wires back from pins 1, 3, 5, 8 and 9 of J8 (noting the colours) to the transformer. The coloured wires should terminate in numbered tags on the transformer. For example, the 'skinny' brown wires to the monitor are terminated on tags 7 and 8 of the transformer (as seen in your photograph).
Disconnect the monitor and J8 from the logic board. Set your multimeter to AC VOLTS (20 ish) and measure the voltages of the following:
Between transformer tags 7 and 8 (brown wires to the monitor) - this should be about 15V to 16V A.C. for a 9" monitor.
The voltage between pins 1 and 3 of J8 should be about 8V A.C.
The voltage between pins 3 and 5 of J8 should be about 8V A.C.
The voltage between pins 1 and 5 of J8 should be about 16V A.C.
The voltage between pins 8 and 9 of J8 should be about 15V A.C.
(Note that it will be much easier to measure the voltages on the tags of the transformer - but for this you will have to identify which pins of J8 go to which transformer tags).
I have an 8032 - but the transformer tag numbers and wire colours appear to be different to your photograph.
You should be able to find the J8 pin-outs on the power supply schematic for your PET and(if you are lucky) the corresponding transformer tag numbers and wire colours.
Reconnect the PET logic board and monitor.
Things should work the same as they did before.
If it was me - I would measure the voltage coming from the 12V regulator of the monitor next. The monitor has a 7812 D.C. voltage regulator IC. You should be able to find this on the schematic and on the monitor PCB layout. I would attach a couple of wires to the PCB with a soldering iron and bring out the 0V (GND) and the +12V regulated supply line (possibly across capacitor C3 (a 47 microfarad 16V capacitor). Your meter should measure 12V. If it is low - then we need to look at why.
You need to be VERY CAREFUL around the monitor. There is 10kV (that is 10,000V) here and it will kill you if you are not careful. After you have measured the +12V voltage rail - I would leave things overnight to discharge before touching the monitor circuit again just to be on the safe side.
If the 12V rail appears to be OK - I would look next to measure the -30V (or -45V) D.C. voltage across the brightness potentiometer (depending upon which version of the monitor you have). Which reminds me - can you point me at the schematics on the internet that match your specific model of PET (just the power supply and monitor should do) - so that we both have a consistent set to refer to.
However - before trying ANYTHING - make sure that you understand thoroughly what I am saying and try to follow it on the schematics. I can't take any responsibility for any additional damage that may occur to your PET or injury to yourself. Also - before attempting any sort of measurements - just have a visual inspection of the components and soldering - trying to look for components that seem 'burnt' or 'stressed' or soldering that does not look right.
Let's take things slowly and make sure you understand the individual steps.
Just a thought - do you know any TV repair men/women in your area? I know they are a dying breed - but if it is your monitor that is at fault; they should be able to fix it for you.
Dave