Oct4 is an embryonic stem cell specific transcription factor. ESCs grown in petri dishes are fed media that contain basic fibroblast growth factor (bFGF) which keeps the ESCs undifferentiated. They examine Oct4 mRNA levels in human ESCs (hESCs), hESCs with exogenous miR-223 (micro RNA 223), hESCs with exogenous miR-223 that do not have bFGF in the media, and hESCs without bFGF but with miR-223 inhibitor. Which technique would allow for measurement of Oct4 mRNA levels

You can use the quantitative Reverse Transcription Polymerase Chain Reaction PCR (RT-qPCR) method in order to analyze the Oct4 mRNA level.  

Explanation:  

RT-PCR is the gold standard technique to analyze the expression levels of one particular messenger RNA (mRNA) molecule. In this technique, the total RNA is firstly extracted and transcribed into complementary DNA (cDNA). Subsequently, the gene sequence of interest is then amplified in a standard PCR technique that uses as control one or more genes in order to measure the relative expression levels of the target mRNA, in this case, for example, the Oct4 mRNA.

it will increase due to the time span