My courtesy light dos not com on when door is open

If the dome lights in your 2009 Cadillac CTS are not turning off, it can be a frustrating issue. Here are some steps to diagnose and potentially fix the problem:
1. **Manually Turn Off the Dome Lights:**
- In most vehicles, including the Cadillac CTS, there's typically a manual switch on or near the dome light itself. Ensure this switch is in the "Off" position.
2. **Check for Open Doors:**
- Make sure all doors, including the trunk and the hood, are completely closed. Sometimes, a door ajar can prevent the dome lights from turning off.
3. **Interior Light Dimmer Control:**
- Check the position of the interior light dimmer control, which is usually located on the headlight switch or dashboard. If it's turned all the way up, it may keep the dome lights on. Adjust it to the lowest setting.
4. **Ignition Switch:**
- Ensure the ignition switch is in the "Off" position. Sometimes, if the ignition switch is left in the "Accessory" position, the dome lights may stay on.
5. **Faulty Door Switches:**
- Each door usually has a door switch that signals the lights to turn on when a door is opened and off when it's closed. One of these switches may be malfunctioning. You can try manually pressing and releasing each door switch to see if the lights respond.
6. **Battery Disconnect and Reconnect:**
- If the above steps don't work, you can disconnect the vehicle's battery for a few minutes and then reconnect it. This can sometimes reset the electrical system and resolve the issue.
7. **Consult the Owner's Manual:**
- Refer to your Cadillac CTS owner's manual for specific information about the dome light controls and troubleshooting tips.
8. **Professional Diagnosis:**
- If the issue persists, it may be related to a more complex electrical problem. In this case, it's advisable to consult a professional mechanic or an auto-electrician to diagnose and repair the problem.
It's important to be patient and methodical when troubleshooting this issue, as it can have various causes. If you're not comfortable working on the electrical system of your vehicle, seeking professional help is a wise choice to avoid further complications.

Check fuses under the hood on driver side

https://www.corvetteforum.com/forums/c6-corvette-general-discussion/1700828-help-rearview-mirror-lights-won-t-turn-off.html
This forum from corvette may shed some light on what to do.

The BCM receives battery voltage from the CTSY/RF TRN fuse in order to supply power to the courtesy lamps. The BCM applies this voltage to the courtesy lamps through the courtesy lamp supply voltage circuits. When any door is opened, the door open switch contacts close and the BCM receives a door-open signal. If the BCM receives a door lock or unlock signal, the BCM will activate the courtesy lamp supply voltage circuit, providing battery voltage to the dome lamps and the left and right footwell courtesy lamps. The courtesy lamps can be manually activated by placing the interior lamp switch in the DOME position. For more detailed information concerning courtesy lamp operation refer to Interior Lighting Systems Description and Operation.
The BCM receives battery voltage from the INT LAMPS fuse in order to supply power to the inadvertent courtesy lamps. Inadvertent power courtesy lamp feature provides a similar function for the interior lamps as a retained accessory power feature. The BCM will energize the courtesy lamps supply voltage circuit when the BCM is in the active state (awake). When the inadvertent power courtesy lamps supply voltage circuit is energized, it supplies voltage to all the interior lights listed below. The BCM will continue to supply voltage to the interior lamps after the ignition is turned OFF so that the interior lamps may be operated.

When the body control module (BCM) is in an active state (awake) it energizes the courtesy lamps supply voltage circuit. The courtesy lamps supply voltage circuit supplies voltage to all of the interior lamps, including the illuminated entry lamps (courtesy lamps). The illuminated entry lamps are turned ON when they receive a battery positive voltage from the courtesy lamp supply voltage circuit of the BCM. When the ignition is in the LOCK position the BCM will apply the courtesy lamp supply voltage circuit under the following conditions:
• The door jamb input transitions to an active state (opening a door).
• The use of the remote keyless entry (RKE) to unlock the side doors.

Do you know how to do automotive electrical circuit testing using a DVOM- digital volt ohm meter and a wiring diagram to pin point testing areas of a circuit ?
Circuit/System Description
The body control module (BCM) applies battery voltage to the courtesy lamps through 2 separately controlled courtesy lamp supply voltage circuits. When any door is opened, the door ajar switch contacts close and the BCM receives a door-ajar input. If the BCM receives a door lock or unlock input or a rear hatch open input, the BCM will activate the courtesy lamp supply voltage circuits, providing battery voltage to all of the instrument panel (I/P) courtesy lamps, dome lamps, rearview mirror lamps, and rear compartment lamps. In the event that any of these lamps were to remain illuminated for more than 20 minutes with the ignition switch in the OFF position, the BCM will deactivate the courtesy lamp control circuits to prevent a total battery discharge condition.

You can view free wiring diagrams here http://www.bbbind.com/free_tsb.html Enter vehicle info. year , make , model and engine ! Under system click on lighting ,then under subsystem click on interior lighting . Click the search button then the blue link. Looking at the first diagram you will see on the top left a fuse box , this B+ for the BCM ,top right is the BCM .You will see three wires coming out the bottom of the BCM , these go to your courtesy lamps , The thing's that look like a open switch inside the BCM , they are transistors or high side drivers . When commanded these supply B+ voltage to the lights . Finding the BCM an testing output voltage would be my first step , before taking the inside of the vehicle apart to find wiring harness to test .

Overhead Console Lamp and Roof Rail Courtesy/Reading Lamps
The courtesy/reading lamp is a duel purpose lamp that can be illuminated two different ways. First, the lamp can be turned ON during courtesy lamp operation as described above. Second, the lamps can be turned ON individually for reading lamp operation by the lamp switch. The BCM supplies battery voltage from the INT LAMPS fuse through the inadvertent power courtesy lamps circuit to the courtesy/reading lamps for reading lamp operation only. Ground for the courtesy lamp module or the roof/rail reading lamps is provided at G402

Courtesy Lamps
When either one of the front doors is opened, ground is applied through the door latch door open switch and the door open switch signal circuit to the door module indicting the door open position. In response to this signal, the door module sends a GMLAN serial data message to the BCM of the door open position. Or, when either one of the rear doors is opened, ground is applied through the door latch door open switch and the rear door open switch signal circuit to the BCM is indicating the rear door open position. Once a door open message or signal received, the BCM will then apply battery voltage from the CTSY/RF TRN fuse through the courtesy lamp supply voltage circuits illuminating the lamps listed below:
• Overhead console lamp
• Roof rail courtesy lamps
• Footwell courtesy lamps The BCM receives battery voltage from the INT LAMPS fuse in order to supply power to the inadvertent courtesy lamps. Inadvertent power courtesy lamp feature provides a similar function for the interior lamps as a retained accessory power feature. The BCM will energize the courtesy lamps supply voltage circuit when the BCM is in the active state (awake). When the inadvertent power courtesy lamps supply voltage circuit is energized, it supplies voltage to all the interior lights listed below. The BCM will continue to supply voltage to the interior lamps after the ignition is turned OFF so that the interior lamps may be operated.
• Overhead Console Reading Lamps
• Roof Rail Reading Lamps
• Vanity Mirror Lamps
• I/P Compartment Lamp
• Rear Compartment Courtesy Lamp
In the event that any of these lamps were to remain illuminated for a period of more than 20 minutes with the ignition switch in the OFF position and no BCM input activity, the BCM will deactivate the inadvertent power supply voltage circuit to prevent a total battery discharge condition. The BCM receives battery voltage from the CTSY/RF TRN fuse in order to supply power to the courtesy lamps. The BCM applies this voltage to the courtesy lamps through the courtesy lamp supply voltage circuits. When any door is opened, the door open switch contacts close and the BCM receives a door-open signal. If the BCM receives a door lock or unlock signal, the BCM will activate the courtesy lamp supply voltage circuit, providing battery voltage to the dome lamps and the left and right footwell courtesy lamps. The courtesy lamps can be manually activated by placing the interior lamp switch in the DOME position. For more detailed information concerning courtesy lamp operation refer to Interior Lighting Systems Description and Operation.

Courtesy Lights
The courtesy lights come on under the following conditions:
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Any door is opened.

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The interior lamp switch is in the ON position.

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The keyless entry transmitter unlocks the doors (illuminated entry).

When the key is removed from the ignition switch with all of the doors closed, the courtesy lights light for approximately 25 seconds before fading out.
Battery positive voltage (B+) is applied at all times through the IP fuse block CTSY LP/PWR MIR fuse 6A via CKT 640 (ORN) through the body control module (MALL) to the courtesy lights via CKT 1732 (ORN). The MALL provides a ground through CKT 156 (WHT) for the courtesy lights under the following conditions:
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The MALL receives an input from a door being opened.

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The MALL receives an input from the interior lamp being turned ON.

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The MALL receives an input from the doors being unlocked using the keyless entry transmitter.

The courtesy lamps then light.
When the courtesy lights are activated by opening and closing a door, the courtesy lights will remain on for approximately 25 seconds before fading out after all of the doors are closed. If the ignition switch is turned to START or RUN within the 25 seconds, the lights immediately begin to fade out. If a door is left open, the MALL removes battery positive voltage (B+) to the courtesy lights after 10 minutes. When the courtesy lights are activated by the interior lamp switch, the lights remain on as long as the interior lamp switch is in the ON position. When the courtesy lights are activated by unlocking the doors using the keyless entry transmitter, the lights remain on for approximately 40 seconds. If the doors are locked using the keyless entry transmitter within the 40 seconds, the lights immediately begin to fade out.
Body Control Module (MALL) Diagnostic Mode
For a detailed explanation of the MALL module diagnostic mode, refer to Body Control Module Cell 51.
What interior light ? Dome , courtesy lights etc....They don't work with switch, keyless entry , door opening ?
What

Page 116 and 117 of the manual explain that there's a two-position slide switch on the map light, DOOR and OFF. The back light has three positions, ON, Door Activated, and OFF.

Pages 116 and 117:
http://techinfo.honda.com/rjanisis/pubs/om/ACC080/ACC0808OM.pdf

Check and see if the lights come on with the passenger door, if so you have a defective courtesy light switch inside the door.

Check to see if your "Dome Overide" is activated