In a circle, the measure of an inscribed angle is half the measure of the arc that it intercepts. In triangle LMN, angle N is inscribed, and it intercepts arc MN.
Given that angle L measures 37°, the opposite arc, which is arc MN, can be found using the following relationship:
\[ \text{Arc measure of } MN = 2 \times \text{measure of angle N} \]
Since angle L (37°) is at vertex L, we calculate the measure of angle N as follows:
\[ \text{The angles in triangle LMN sum up to 180°} \] \[ \text{If we denote angle M as } m \text{ and angle N as } n, \] \[ 37° + m + n = 180° \] \[ m + n = 143° \]
To find the inscribed angle N, let's use the relationship that angle N intercepts arc LM. Since LM has a measure equal to \( 180° \) (diameter):
\[ \text{Arc measure of } LM = 180° \implies m = 90° \text{ (by complementary angles)} \] \[ n = 143° - 90° = 53° \]
Now, we want to find the arc MN which is intercepted by angle N:
\[ \text{Arc measure of } MN = 2 \times 53° = 106° \]
Thus, the arc measure of MN is
\[ \boxed{106°} \]